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The Trouble With Touching: A Problem In 's Continuity Theory

by

Sammy T. Jakubowicz

Graduate Program In

Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy

Faculty of Graduate Studies The University of Western Ontario London, Ontario July 1999

O Sammy T. Jakubowicz 2999 National Library Bibliotheque nationale 1*1 of Canada du Canada Acquisitions and Acquisitions et Bibliographic Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON K1A ON4 Ottawa ON KIA ON4 Canada Canada Your Me Vofre referame

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The author has granted a non- L7auteura accorde une licence non exclusive licence allowing the exclusive pennettant a la National Library of Canada to Bibliotheque nationale du Canada de reproduce, loan, dstribute or sell reproduire, preter, distribuer ou copies of this thesis in microform, vendre des copies de cette these sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format eiectronique.

The author retains ownerslup of the L'auteur conserve la propriete du copyright in this thesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts fi-om it Ni la these ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent Ctre imprimes reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract

Aristotle's rests upon the foundation of the continuum. This entails

understanding to be infinitely divisible. However, when space is understood in this

way we find a of . In particular, this work deals with the difficulties posed

by the problem of touching in Aristotle's continuum. It would seem that, given his

understanding of space, it would not be possible for any two objects to come into contact.

And this would apply not only to physical situations, but to geometricaf ones as well.

In order to solve this difficulty, or, more accurately, to see how it might have been

possible for Aristotle himself to have solved it, a number of related are examined. The early stages of the examination focus primarily on the geometry of the situation and explore

the of points and boundaries. After acquiring a better understanding of these and of what they entail, we are able to move to the physical side of the problem and examine such concepts as intelligible , mixing, and actuality. When taken together, these allow us to do two things. First, we are more easily able to see the relation between the physical and the geometrical concepts and difficulties. More importantly, we are able to see how the initial problem of touching, as well as the variations on the problem, could have been solved by Aristotle within his own framework.

Keywords: Aristotle, touching, point, boundary, limit, geometry, intelligible matter, mixing, actuality, potentiality, place, continuum, space, unit, tangent, history of , Appropriately enough, given Aristotle's views, a thesis is not written in a .

There are a great many people who should be acknowledged for the parts they played in bringing this work to completion.

My supervisor, John Thorp, deserves a great deal of credit for guiding me through the process of writing this thesis. My finding this topic is a direct result of having taken, and having enjoyed, his courses. Without his advice and constructive criticism this would have been a far different - and probably far less interesting - work.

I would also like to thank the members of my advisory and defence committees -

John Bell, Lome Falkenstein, Paul Potter, hing Block, and John Magee - for their comments on the thesis. Thanks are also due to John Scott who, at the 1997 meeting of the

CPA, commented on a very early draft of the chapter on points. Their comments were helpful and appreciated.

Since my arrival in this department I have felt welcome, and this has made my work all the easier. This is in no small part due to the people who REALLY run the show - Andrea

Purvis, Julia Mcdonald, Glenda Ogilvie, and Ella Young. Credit should also be given to the friends I have made while living in London. Thank you all for putting up with my idiosyncrasies.

And last, though by no means least, are the members of my family. None of this work would have been possible had it not been for the support and encouragement they gave me from an early age. I place the blame for my love of learning squarely on your shoulders, and I thank you for giving me the blame to place. Table of Contents

Page .. Certificate of Examination ll .. . Abstract 111 Ac know Iedgements iv Table of Contents v A Note on Texts vii

Chapter 1 - Introduction 1 The Problems 2 The Method 8

Chapter 2 - Points 11 The Traditional Interpretation of Points: The EuclideanKartesian View 13 What Is a Unit? - A Digression Into How Units Relate to Points 16 Problems With the Traditional Interpretation of Points I. The Place of Points 26 2. The Size of Points 29 3. Points in Contact 3 1 A Different Interpretation: Ideal Versus Real 35 The Advantages of this Interpretation 47

Chapter 3 - Boundaries 53 What is a Boundary? 55 The Problem of Coincident Boundaries 66 The Problem of a Shared Boundary 70 Aquinas' Solution - Touching Boundaries vs. No Boundaries 76 Another Solution - Boundaries in Contact vs. Mixed Boundaries 83 The Missing Piece of the Puzzle: Mixing 87

Chapter 4 - Geometrical Objects and Intelligible Matter 9 1 Why the of Mixing Is Important to the Project 92 Aristotle's Views on Mixing 94 Difficulties with Aristotle's View 1. The Problem of Coincident Bodies 97 2. The Problem of Instantaneous Change 100 3. The Problem of Generation 100 4. The Problem of Geometricd Mixing 102 Intelligible Matter - Solving the Problem of Geometrical Mixing 103 The Difficulty of Making the Transition from Physical to Geometrical Objects 1 15 Solving the Other Difficulties 120 Chapter 5 - Actuality and Potentiality Connecting Intelligible Matter to the Actual/Potential Distinction Explaining Actuality and Potentiality Actualization and Geometry Actualizing Potentials Solving the Problems with Boundaries Solving the Problems of Coincident Boundaries, Instantaneous Change, and Generation

Chapter 6 - Conclusion Review: What Has Been Uncovered Thus Far Solving the Original Problems 1. Basic Tangency 2. Touching at Boundaries 3. Physical Tangency and Abstraction The Significance of the Findings

Bibliography

Vita A Note on Texts

References are made throughout this dissertation to the works of Aristotle. Unless

otherwise indicated, all English translations of the works of AristotIe have been taken from

Aristotle. The Complete Works of Aristotle. Ed. Jonathan Barnes. 2 vols. Princeton: Princeton University Press, 1995.

The about the Greek texts I have used can be found in the bibliography.

vii Cha~ter1 - Introduction

The Problems There are a number of important problems which arise when we Look closely at the of continuity as it is presented in Aristotle's writings. This section introduces four difficulties related to the problem of touching in a continuum: the problems of geometrical and mathematical tangency, the problem of contact at boundaries, and the problem of abstraction. These need to be addressed in order to understand how Aristotle understood the of contact in both a geometrical and a physical context.

The Method This section serves as a rough guide to the manner in which the project will proceed. In particular, the objectives of the ensuing chapters are mentioned in order better to place them in the broader context of the project at large. 2

The Problems

Much of Aristotle's physics rests on the foundation of what is called 'continuum

theory'. This theory consists in seeing matter, space, and as divisible ad infiniturn.

However, there is a difficulty with this theory in that it cannot coherently deal with the idea

of two objects touching one another. It may be that Aristotle himself was aware of this problem and this can be seen through his various attempts in his works to solve related problems. These attempts, though, often do as much to complicate as they do to solve them. In fact, at the entire exercise seems like a neurotic effort on Aristotle's part to save his theory.

Consider the following diagram.

If two circles touch at a single point, call this point C, they are understood as tangent to each other at that point; we standardly anaiyse this mathematical situation as one in which the two circles share a single point, and this analysis gives us little trouble.

This problem becomes more complicated when we try to see how it would work in a physical setting. Assume that the circles represent two material spheres which touch one another and which exist as continuous bodies in a continuous space of the type for which

Aristotle argues so vehemently. What analysis can be made of the point at which the spheres 3 come into contact? When the problem dealt only with nonmaterial elements such as Iines and circles we were not troubled by saying that two shapes shared a point. However, unlike the situation where the circles are mere geometrical entities, in the material version of the problem we would not want to say that this point is in both spheres. If it were, then the two objects would intermingle and lose their discreteness; we would have a single , not two objects touching one another. We might say that the spheres could be divided at the point of contact, but this would not work either. To do so would be to say that points are divisible and, therefore, have size. However. Aristotle seems to tell us that points are indivisible, so this solution is unacceptable. Nor would we want to say that the point C is in sphere A alone, nor in sphere B alone, as we want it to be a point of contact. If the point is only in one of the spheres, then there is no contact. Nor would we want to say that the point is in neither sphere. After all, if the point is in neither, then it would seem that the spheres would not be touching: there would be something between them. This seems to eliminate all possibilities, and leads to the striking concIusion that in a continuum physics like that of Aristotle two objects cannot touch one another. But objects in the red world do touch one another, so the physics must be wrong.

In fact, this question about two objects contacting one another at a point is only a small part of the problem. If mere points present us with such a difficulty, consider the same issue in a higher . By this I mean the situation of objects touching each other at lines or planes rather than merely at points. Richard Sharvy gives a good description of this problem as it wouId occur with figures touching at a line.

..,if the Oregon-Washington border is part of each state, then those states have points in common, which seems odd; if it is part of neither, then a road crossing from one to the other has points that are in neither, which also seems odd; and it would be odd for that border to belong to one of these states rather than the other. '

Sharvy describes this problem as a 'real "don't care",' and claims that by sticking to a definition of points defined as limits this sort of problem can safely be ignored? However, it seems to me that if we are to understand continuity theory and all the implications it has then we must care about such a problem. What this problem shows is that there is a difficulty with the idea of touching in a continuum. While it may seem easy to solve when considered in the abstract, as it did earlier with the problem of the two tangent circles, it is obviously something that needs to be looked at more carefully, especially when considered as a physical circumstance and not merely as a mathematical puzzle. Sharvy's suggestion that we consider points as limits in order to avoid such problems leaves much to be desired.

It does not explain why we should ignore else Aristotle wrote on the subject, as

Sharvy's position would require, nor does it fully solve all of the difficulties this type of problem suggests, as will be seen.

While the problem of touching can clearly continue to be extended to higher , the difficulties involved with it do not end with this important, yet somewhat esoteric problem. A large part of what is at issue here is exactly that which Sharvy says will help us avoid the problem. Specifically, we find even more problems when we examine the idea of boundaries and their application in the physicai world. By this I mean that as students of natural philosophy, we can make a distinction between those things which exist only in , such as the objects of mathematics and geometry, and those things which make up 5 the physical, sensible, changeable world around us. When we regard the world as something physical as opposed to mathematical, we see objects which are in contact with one another.

In fact, under an Aristotelian conception every object must be in contact with some other object. If this were not the case, then we would have a situation in which some object was surrounded by a void. It is clear, given his attitude towards the of the void, that

Aristotle would not admit this as possible. Not only do we have objects in contact with one another, but we also have some objects which are continuous. If we consider the earlier problems with this in , then we realize that there is a much larger problem to be dealt with. If we examine two contiguous objects, say a bronze sphere immersed in water, we are again faced with the problem of touching. In order to avoid accusations of allowing empty space, we must have contact of some kind. However, if any part of the boundary between the water and the bronze is shared then we face a difficult situation. How, ultimately, are we to distinguish between the water and the bronze? At best, we have a mixture of the two substances at that problematic region which we identified as the boundary. And even assuming that we could eventually make such a distinction, how could we distinguish between the boundary and the bronze? Even worse, since it would seem that a boundary must itself be in contact with part of that which it limits, the impurities in the boundary will make their way into the bounded body. It seems that we would be left with a regression which would leave us with no part of the bronze sphere made of bronze alone, and no part of the water made of water alone. There would always be some corruption from the substance which was touched. This is hardly satisfactory. The implication is that , not even the basic eIements of which is made, could be 'pure'. Taken to an extreme, 6 this leads us to a picture of the world not unlike that given by Empedocles.

The idea that shared boundaries seem to lead to an Empedoclean picture should probably be explained more clearly. Aristotle himself tells us that Empedocles had the four basic material elements in his view of the world - earth, air, fire, and water? However, we also know that he had two guiding elements - Love (Qlh6~q5)and Strife (NE~Ko<).It is through these guiding elements that one comes to be from many and vice versa." I interpret this as that when Strife reigns over the world, dlthe elements are separate from one another, but when Love is the dominant , then everything that exists is mixed together in one great Dali-esque whole. This picture, namely the situation where everything is adulterated with everything else, would seem to be the picture we would have to accept given the idea of shared boundaries as it has been presented. Of course, were we to accept such a world view, then we would have no pure substances which could be adulterated. While this would solve the problem of mixing at a boundary, I do not think that it Aristotle understood the world in this way. Thus, it seems that if we are to have pure substances which touch, then we are left with this difficulty.

Until now, I have been focussing on the problems of contact in the physical world and treating geometrical situations as though they present us with no troubles. Lest there be any doubt, the mathematical world as understood by Aristotle is also threatened by the problems discussed above. Though I have been suggesting that we have no problem with the idea that two circles can be said to share a point, or that we have no difficulty with the notion of tangency, this is not altogether correct. Recall that we should be attempting to look at the situation from an Aristotelian perspective, not from our own. While we may understand 7 these concepts, the problems inherent in continuity theory must plague Aristotle's understanding of geometry. This is because we are told that the objects of geometry are found through abstraction. However. if this is the case, then any problem we encounter with boundaries in the physical world must also show up in the realm of mathematics. Thus, suppose I am unsure about the location of the boundary between two objects and I "abstract" to find the boundary: if I say that the boundary is shared at even a single point, I may be forced to resign myself to saying that the physical objects from which I have abstracted are indeed mixed. If, however, I deny such an idea, then I can never abstract to a geometrical situation which involves tangency or some related notion. I might still be able to conceive of such a situation, but I would be forced to say that this could never actually happen.

Given all these problems, it would appear at first that in the problem of contact. a fairly obvious aspect of continuity was overlooked by Aristotle. This would be strange given the depth and sophistication seen in his various discussions of the continuum, such as his discussions of the first and last moment of change in the physica5 and of his theory of touching presented in the De ~nima.~However, I think that a deeper reading of the corpus shows that Aristotle was aware of some sort of problem with his theory when it tried to deal with two objects coming into contact with one another, though he could not quite isolate this problem. Aristotle's various attempts to find a solution paint a picture of the unease in his mind caused by the knowledge that something is not quite working here; indeed, these solutions sometimes seem less like solid conjecture than they do like grasping at straws. My thesis seeks to expound, to organize, and to consider Aristotle's twistings and turnings as he confronted the trouble with touching. 8

The Method

What should be made clear here is that the solutions I intend to present to these problems are not necessarily those which Aristotle himself would have presented. As I have suggested, Aristotle seems to have been aware of some of the difficulties inherent in his system. but I do not believe that he had as clear a view as I am about to present. This does not mean that there was no awareness of the problems. As will be seen, most if not all ofthe families of solutions which will be presented can be directly derived from Aristotle's understanding of both mathematics and physics. To go beyond this is, I think, wrong. My in this work is not to show that Aristotle was wrong. We already know that he was mistaken about a great many things, and to point out all the flaws in Aristotelian mathematics and science would really only serve to show that we can generally give better answers after working on a problem for two thousand years. Instead, my intention is to show that despite the apparent difficulties, Aristotle's system can indeed work as a coherent whole. That is to say that what I plan to accomplish is to show that Aristotle's geometry and mathematics, while not what we generally accept as correct, was not as flawed as the difficulties above seem to suggest. I will illustrate this by showing that Aristotle had the tools at his disposal, and in fact often made relevant comments and observations, so as to be able to solve difficulties such as those presented above. It should be emphasized, however, that although

I will present solutions of which I think Aristotle would have approved, I do not claim that he himself gave such solutions. This is especially important to bear in mind in the final chapters of this thesis, as at times the line distinguishing between what Aristotle actually wrote and what I believe might solve the problems raised becomes blurred. 9

In order to understand how Aristotle dealt, or at least could have dealt, with these issues a number of concepts must be examined. First, there is the notion which seems to have opened this can of worms - points. Points, as they are commonly dedt with in modem commentaries on Aristotle, seem to be exactly those things which we think of when we talk of points. That is to say that they seem to be geometrical entities lacking both size and shape.

While they are represented graphically by dots, they are not even that substantial. However, what I will show is that the conception with which Aristotle presents us is far more complex than this. In fact, as I interpret Aristotle, he seems to give us two types of points - one real and the other ideal. It is the confusion between these related though distinct types of points which cause many of the problems we find when we try to assess the concept of contact.

After dealing with points to the extent that this is possible, I will move, as I did in the various problems I discussed earlier, to a higher dimension. That is, I will deal with objects which are bounded. As has already been mentioned, points can be defined as limits.

Therefore, by thinking of points as the limiting case of boundaries, the discussion of boundaries will hopefully allow us to solve the unanswered problems from the discussion of points. Several families of solutions will be examined, though further information will be required before these solutions will be viable.

The next two chapters will present the information required to make use of the earlier flawed solutions. First, I will make a stronger connection between the realms of mathematics and physics. This connection will be made using the concept of intelligible matter. By understanding intelligible matter as the substratum for other matter, we can better understand the concept of mixing. This, in turn, will allow us to solve some, though not all, of the 10 difficulties we will have encountered in the earlier chapters. Those difficulties which intelligible matter will not solve by itself will be seen to resolve themselves once we have a better understanding of the final concepts to be introduced into the mix, namely those of actuality and potentiality. Once explained, these concepts will allow us to return to the troublesome solutions from earlier chapters and show how, using only those concepts available to Aristotle, the problems associated with touching in a continuum could be solved.

1. Kchard Sharvy, "Aristotle on Mixtures", The Journal of Philosophy, LXXX ( 1983),449.

2. ibid.

4. G.S. Kirk, J.E. Raven, M. Schofield, The Presocratic Philosophers (2nd edition), (Cambridge University Press, 1983), 287-288. Chapter 2 - Points

The Traditional Interpretation of Aristotle's Idea of Points: The EuclideadCartesian View This section begins with an explanation of what a point was supposed to be by both Euclid and Descartes. In particular, we see that points are seen as entities without dimension or sensible qualities. Following this, it is shown that Aristotle's conception of a point differs from these later conceptions at least in the that points seem to be connected explicitly to position.

What Is a Unit? - A Digression Into How Units Relate to Points Units are defined as points without position; a digression is made in order to understand what exactly is meant by this phrase. Units are seen to come in two varieties: the common unit, which is essentially a measure of quantity. and the philosophical unit, which conforms in all ways to the common unit with the added condition that each philosophical unit be exactIy the same size. Further, we see that the indivisibility of units is not absolute. Units can be divided, but they then no longer conform to the previously defined unit.

Problems with the Traditional Interpretation of Points There are three main problems which arise if we try to read Aristotle's texts with a Cartesian or Euclidean conception of points in mind. First, we tend to define a point in terms of position, but Aristotle says that a point can change place. Second, we say that a point has no dimension, yet Aristotle sometimes speaks of points as though they have size. Third, we have difficulties with the idea of points coming into contact with one another, but at times Aristotle seems untroubled by this idea. This section clarifies these difficulties.

A Different Interpretation: Ideal Versus Red A possible solution is presented here which may ease the difficulties from the previous section. Rather than making the claim that Aristotle had a clear understanding of points, this section puts forth the idea that AristotIe had two distinct ways of thinking about them. On the one hand, they were thought of as ideal objects, as geometers tend to think of them. On the other hand, points were more easily understood as something akin to dots or even .

The Advantages of This Intemretation Here I explore some of the expected benefits of this double way of thinking about points. We see that there are some advantages, but also some unanswered difficulties which require more analysis. The most pronounced among these is the problem raised earlier, of points corning into contact. This prompts a move to the related topic of boundaries. 12

Traditionally, though not universally, the position taken by modem commentators

on Aristotle, such as Ross and Heath, seems to indicate that we can simply map our own

understanding of mathematics onto ~ristot1e's.l That is, we are simply to understand his

remarks on points as references to sizeless, dimensionless geometrical entities. I think this

is wrong. I think it would be better to understand Aristotle as grappling with and groping

towards the idea of points, and as not altogether settled about what they are. This is especially important to the question at hand.

The problems discussed in the previous chapter have essentially dealt with touching in two distinct ways - physical and geometrical. The former is seemingly more complex.

While touching in a mathematical sense involves consideration of geometrical objects and how they exist and relate to one another, touching in a physical sense seems to involve all the considerations present in geometrical cases and more. For example, a geometer routinely considers shapes in the absence of anything else. Thus, a sphere can be examined without worrying about what it is that surrounds that sphere. However, if the geometer then considers an orange qua sphere, leaving aside for the time being any womes about the imperfections that cause the orange to deviate from the shape of a perfect sphere, we must also consider the surrounding medium. The orange does not, after all, exist in a vacuum.

It may, for example, be surrounded by air or water. Unlike the case of purely geometrical objects we must determine how these different physical things behave at their boundaries.

Given the relative complexity of touching in a physical sense when compared to the mathematical sense, it would seem prudent to start by examining what is meant when we say, or more preciseIy what is meant when Aristotle says, that we have touching between two 13

objects and these objects are geometric& in nature. And in order to simplify the matter, it

would be best to start by examining the simplest of all geometrical objects, the point.

The Traditional Intermetation of Aristotle's Idea of Points: The Euclidean/Cartesian View

The earliest definition of points in antiquity of which we know is not Aristotle's, but

rather appears to have been given by the ~~tha~oreans.~Our concern, though, is with

Aristotle's struggle with the concept and the andysis he gave. Traditionally, as I have

mentioned, Aristotle is thought of as understanding points as dimensionless entities. If this

tradition is correct, though, it would mean that Aristotle was not the first, nor was he the last

person to understand points in this way. For example, although he worked after Aristotle and does not go further than a simple definition, Euclid tells us what a point is so that he can

proceed with various propositions in his Elements. There he claims that

Xqpei6v konv, 06 pEpoq 0686~.

A point is that which has no part.3

Leaving Aristotle aside for the moment, this is not in fact, the earliest definition of this sort.

As will be shown later, definitions of a sirni1a.r type were known before Euclid. However, it is Euclid's treatise which formed the cornerstone of geometry for over two thousand years.

In the seventeenth century, Rene Descartes came up with the idea of using the lengths of lines to define points, and this idea is what allows us to use ordered pairs of to define the position of any point on a .4 The fxst sentence of his Geometry reads

All the problems of geometry can easily be reduced to such terms that thereafter we need to know only the length of certain straight lines in order to construct them.' This idea, thinking of a point as an ordered pair of numbers rather than an object of some sort, is quite different from the idea of point which peaked with Euclid's definition in the

Elements. Despite some problems with Euclid's definition$ these two views of points, the first purely geometrical, the other forming a link of sorts between geometry and algebra, have been generally accepted for centuries. It is for this reason that I call these views of points, which form the basis of the modem understanding which is, tun, all to often mapped onto Aristotle, 'Euclidean' and 'Cartesian'.

As I have said, there were definitions and discussions of points prior to that given by

Euclid. Consider the following - although Aristotle occasionally uses the same term for

'point' later used by Euclid, namely 'oqpiov', the word Aristotle more commonly uses for point is 'cmypfi'. In Historia Animnlium, he writes

... the heart appears, like a speck of blood in the white of an egg. This point beats and moves as though endowed with ...'

Clearly this is not a ottyptj or a oqp~iov,a point, in the modem or even the ancient geometrical sense. It is rather a mark or a jot. C. P. Jones, in a paper about tattooing in antiquity gives a brief etymology of the word onypj,which is the term which primarily will occupy our exploration.

The verb oriC~~vmeans 'to prick', and is related to the English 'sting', 'stick', 'stitch', to the German 'stechen' ('prick', 'puncture'), 'Stick' ('prick', 'sting', 'puncture'), 'sticken' ('embroider'). Stigma first appears in Greek with reference to the spots of snakes, and it continues to bear the meaning of 'dot', 'mark', 'welt', which is one of the reasons why it is sometimes difficult to be sure that tattoo- and not brand-marks are in question. Yet azi&w is the appropriate word to describe the process of tattooing, and stigma to describe the mark so made; and at least until the Roman imperial period this is the meaning suggested by the texts.'

One might think that the fact that attypj can mean 'dot'. 'mark',and 'welt' besides meaning

'point' is nothing more than a case of outright ambiguity (what Aristotle calls 'homonymy'):

two quite different meanings of a single word, appropriate to different contexts - geometry

on the one hand and the barnyard on the other. Unfortunately, I do not think things are to

be solved quite so easily. The evidence I will consider suggests that Aristotle sometimes had

an understanding of the point in geometry which was not only different from our own, but

also something more akin to . My intention in what follows is to examine several

texts which will show that Aristotle seems to have held two different conceptions of

geometrical points; an ideal conception, which we might see as akin to our modem

understanding, and another less abstract, but very perplexing, conception where points are more Like very small dots.

A good place to begin looking at Aristotlets notion of points is the Melaphysica. In book A, he writes

That which is indivisible in quantity and qua quantity is called a unit if it is not divisible in any dimension and is without position, a point if it is not divisible in any dimension and has position, a line if it is divisible in one dimension, a plane if in two, a body if divisible in quantity in all - i.e. in three - dimension^.^

This seems to establish fairly clearly his basic understanding of points. They are indivisible objects of a geometrical nature which possess position. In fact, a point is intimately tied to position in a way that other geometrical objects are not. In the Physica. during his discussion of place, Aristotle writes

Further, if body has a place and a space clearly so too have surface and the other limits of body; for the same argument will apply to them: where the bounding ptanes of the water were, there will in turn be those of the air. But when we come to a point we cannot make a distinction between it and its place. Hence if the place of a point is not different from the point, no more will that of any of the others be different, and place will not be something different from each of them.''

Thus, if we attempt to conceive of a point without position, we are left without a point to conceive. At best we would be left with a 'unit' (yovci~)which we are repeatedly told throughout the corpus is 'a point without position' (iy8p povh~or~ypfl &0~tbq konv)."

In a strict translation, we might say that we have in a povoiq a point without position or without place. But what an odd idea this is! At the very least it shows us that Aristotle's array of geometrical concepts does not map directly onto ours.

What Is a Unit? - A Digression Into How Units Relate to Points

At this juncture, it is necessary to make a digression from our project of examining points. After all, if it is possible that what we have just seen is true, namely that we are left with a unit when we remove position from a point, then we might say that units are more basic than points. If so, then since the project as it was defined earlier was to examine the simplest of geometrical entities and how touching works for them, it would make sense to examine the properties of units and to think about how they can touch. I call this a digression and not a fresh start since to consider units as the simplest geometrical entities would be to attribute to them properties which I do not believe belong to units. I say this because traditionally, the unit is thought to be some sort of proto-mathematical entity. In fact, both Heath and Evans make the claim that the earlier Greek thinkers considered units to be nothing more than just the building blocks of numbers." Heath writes

According to Iamblichus, Thymaridas (an ancient Pythagorean, probably not later than 's time) defined a unit as 'limiting quantity' (rcepaivouoar xoobrq~) or, as we might say, 'limit of fewness', while some Pythagoreans called it 'the confine between number and parts', i.e. that which separates multiples and submultiples. l3

I do not know whether AristotIe would have accepted exactly these definitions, but it is not inconceivable given his approval of the doctrines of the later Pythagoreans with respect to definitions of the base geometrical objects." What is clear about Aristotle's conception of unit is that it is the of number.'* We find him asserting in the that

...measure is that by which quantity is known; and quantity qua quantity is known either by a 'one' or by a number, and all number is known by a 'one'. Therefore all quantity qua quantity is known by the one, and that by which quantities are primarily known is the one itself; and so the one is the starting-point of number qua number. l6

Annas interprets this as saying that the unit, or whatever is counted as one, as Aristotle suggests here, is the measure of number."

Given the traditional use of units as essentially arithmetical entities, or at the very least proto-arithmetical entities, we must wonder what their connection is to points. Recall that this digression was prompted by the assertion that units are points without position. Thus we must wonder whether points for Aristotle are completely different from units, losing the properties of units and taking on different ones when they take on position, or whether points are simply units with an added feature. The answer would seem to be that Aristotle, following the Pythagoreans, believed the latter. The Pythagoreans connected the unit in arithmetic and the point in geometry by saying that the unit is a 'point without position'

(ostypfi &0arog), and a point is 'a unit having position' (povh~0dow &~ouocr).'~While I have only given one instance from the Metaphysica, Aristotle often repeats these phrases when he discusses points and units. Annas would have us go beyond Heath's assertion that the notion of point without position is derived from Pythagorean doctrine. She believes that this is not simply a Pythagorean idea, but that it derives from a closer source. namely Plato arid the lectures he delivered in the ~caderny.'~This, despite the fact that we are told that

Plato objected to the definition of point as a 'unit having position'.20 Annas finher claims that the relation between points and units is really the relation between the theoretical and the practical. She refers to the Philebus to make her case:

Socrates: To take first the numbering or arithmetic, ought we not to distinguish between that of the ordinary man and that of the philosopher? Protarchus: On what principle, may I ask, is this discrimination of two arithrnetics to be based? Socrates: There is an important mark of difference, Protarchus. The ordinary mathematician, surely, operates with unequal units; his 'two' may be two armies or two cows or two anythings from the smallest thing in the world to the biggest, while the philosopher will have nothing to do with him, unless he consents to make every single instance of his unit precisely equal to every other of its infinite number of instances. Protarchus: Certainly you are right in speaking of an important distinction among those who concern themselves with number, which justifies the belief that there are two arithrneti~s.~' 19

What this suggests is that Plato thought of units as the mathematical basis for the common person. Thus, a farmer couId count livestock by assigning the designation 'unit' to each animal. Likewise, that same farmer could count only oxen by claiming that each ox would count as a unit of oxen and that a goat would not be such a unit. In this way, the designation

'unit' would be precise enough for common use, but not precise enough for the philosopher, who, according to Plato, would require all units to be of an equal, if not defined, size. Thus,

Annas notes, we have two ways of interpreting units.22 First we have the common way, which involves a measure of heterogeneity, such that one ox and one army are both examples of one unit despite the differences in size. As well, there is also the manner of the philosopher, which is to insist upon units which are in no way dissimilar. This unit is to count as one, to be one, and to be indivisible. Both views in some way consider the unit as the budding block for numbers. A number is the number of something. Thus, when we are counting, anything being counted which cannot be divided without removing something countable is a unit. For example, when we are counting sheep, a singIe sheep composes a unit. We cannot remove part of a sheep and still be left with the basic countable unit. The difference between the 'common' unit and the 'philosophical' unit is what gives rise to the difference between transitive counting, such as 'one cow, two cows, three cows', and intransitive counting, such as 'one, two, three'.23 Put another way, transitive counting allows us to take any herd of cows and to count them in any order. Intransitive counting is akin to painting numbers on the sides of the cows and insisting that they always be counted in the order dictated by the numbers.

RegardIess of the type of unit we are considering, it is dear that the unit qua unit is indivisible. According to Annas, this indivisibility, be it of the type belonging to the common unit or the philosophical unit, is what allows us to use units as a unit of measurement and the basis of number.

The unit of measurement is what is taken as indivisible for the purposes of measurement, and similarly the unit in arithmetic is what is taken to be indivisible for the purposes of counting or computation. Thus the problem of the indivisibility of the mathematicians' unit is solved without Plato's postulation of perfect pure and indivisible units. The mathematicians' unit is just an ordinary regarded as indivisible for counting; it is not a different sort of object altogether. This characterization of 'unit' appears elsewhere in Aristotle, and when he says that mathematicians posit the unit as in every way indivisible he is making a great advance on the 'point without position' characterization."

One important thing to note here is that although Annas refers to the unit of measurement being indivisible for the purposes of measurement, she is probably not suggesting that

Aristotle was thinking of points as a means of measure. While we are told that Plato held the view that points were mere geometrical fictions and that they were at most 'indivisible lines',25we also know that Mstotle is quite clear about the impossibility of any number of points being put together to make a line, which we might class as the simplest of measurable objects. Evans describes this difference between the use of units and the use of points.

For [Aristotle], point and unit are each indivisibles, and they are distinguishable from one another only insofar as the point has position. So too, the movement of the point generates a line, while the "movement" of the unit, in counting, generates number. However, it is doubtful whether Aristotle saw any redly significant relationship between geometry and arithmetic: he was concerned, primarily, to the difference between the two. Thus he notes that magnitude is divisible into continuous parts, whereas number is divisible into non-continuous parts. He is also aware, of course that the point is not a measure of the line, since a plurality of points does not constitute a line, whereas the unit is the measure of number.26

What this tells us is that a unit can be used to measure number in a very specific sense. Namely, units are used for counting and thus form the basis or measure of number. We can take any whole number, divide it into a group of units, and by this means measure the number. Points, on the other hand, cannot be put together in such a way as to form a continuous magnitude. This means that a point cannot be used to measure a line. This does not preclude our thinking of points as units since, if we foollow the rules out in the passage from the Philebus, they fall under the category of what I have termed philosophical units.

That is to say that, with the possible exception of spatial location - an attribute not relevant to consideration of units - every instance of a point is identical with every other instance.

The idea that Aristotle believed that units were indivisible is not simply a matter of making an inference based upon Plato's thought. It is confirmed by Aristotle in the

Metaphysics where he writes

For all number means a quantity, and so does unit, unless it means merely a measure or the indivisible in quantity.'7

Aristotle presents us here with a . We can either accept a unit as an indivisible measure, which seems to be what Annas and Heath believe, or a unit can simply be a quantity. Given the way this statement is presented, I assume that Aristotle did not necessarily know that units were indivisible in the ways that I have presented. Ross's commentary on this passage is helpful. He writes

The manuscript reading seems to mean that if the unit is not some finite measure it is the limiting case of quantity - infinitesimal quantity; it is always quantity of some kind. But [there are some textual problems and comments from Alexander which] suggest[s] the reading ~i fl phpov KQ~56 K~Z& KTA. 'The unit indicates a quantity unless it means a measure or what is indivisible in quantity.' 'Measure' and 'indivisible in quantity' are not, indeed, absoiute synonyms, for a measure may be an 'indivisible in ', but ti5 &v is a measure of quantity primarily.2s

It would seem then that we are on reasonably steady ground in attributing such indecision about the indivisibility of units to Aristotle. Consider that if Ross's reading is correct, we have a situation in which the limiting case of a unit is a quantum - an infinitesimal quantity to be sure, but still a quantity. If we are going to say that Aristotle accepts the 'point without position' definition of unit, and I believe we have good reason for doing so, then this raises some difficulties when we introduce the modem notion of a point. Recall that we think of a geometrical point as a sizeless, massless entity. Yet, if we accept that a unit has some size and that when given position we have more than a unit - specifically we have a point - we must then accept that one of the properties of a point must be size. When we remember that

Aristotelian thought does not easily accept such a notion - the treatise De Lineis

Znsecabilibus having been written for the purpose of discrediting the similar view that points are indivisible lines - we can understand why there would have been such confusion on

Aristotle's part.

A further possible lack of clarity in Aristotle may be at+tributableto his idea that certain quantities, such as units, must be assumed, while the rest are deduced.

I call in each genus those which it is not possible to prove to be. Now both what the primitives and what the things dependent on them signify is assumed; but that they are must be assumed for the principles and proved for the rest - e.g. we must assume what a unit or what straight and triangle signify, and that the unit and magnitude are; but we must prove that 23

the others are.29

The other possibility, and in my view the more reasonable one, stems from the idea seen

earlier that a unit is the measure of number dong with the idea that units must be of the same

kind. hnas claims that what is meant by 'unit' here is whatever is counted as 'one'. Prior

to measuring, we must have a unit of meas~rernent.~~As well, lengths are measured in units

of length, weights of weight. etc? A unit is regarded as indivisible for counting, though it

may be divisible in other arenas.32 For example, we see in Metaphysicu A that Aristotle

writes

But the one is not the same in all classes. For here it is a quartertone, and there it is the vowel or the consonant; and there is another unit of weight and another of movement. But everywhere the one is indivisible either in quantity or in kind."

Since we have different units depending upon what is being measured, we can say that the units used do not differ from one another, though the things being measured may differ. In this way, we can account for Aristotle's view about why it is that numbers have differentiae, but units do not.

The idea that units have no differentiae, which in a purely spatial sense can be interpreted as their having no parts, ties in quite neatly with the corresponding idea which applies to points. In particular, we see the assertion that points have no differentiae in the pseudonymous, though clearly Aristotelian, text De Lineis where it is said that Again, whenever one thing is contiguous with another, the contact is either whole-with-whole, or part-with-part, or whole-with-part. But the point is without parts. Hence the contact of point with point must be a contact whole-with-whole?

As well, later on in the same book, we find a passage which says

Moreover, how will there any longer be straight and curved lines? For the conjunction of the points in the straight line will not differ in any way from their conjunction in the curved line. For the contact of what is partless with what is partless is contact whole-with-whoIe, and no other mode of contact is possible. Since, then, the lines are different, but the conjunction of points is the same, clearly a line will not depend on the conjunction: hence neither will a line consist of points.35

We can take this to support the claim that if two objects are in whole-with-whole contact by virtue of being without parts then those objects have no differentiae in the spatial sense.

Before proceeding further, it might be helpful to survey the situation and see exactly where this leaves us. This discussion was started because it was necessary to determine what units were and how they related to points. We have seen that at the heart of the unit is the concept of quantity. A quantity, however, is not necessarily to be taken to be a uniform quantity. Thus, we may have what may be termed 'common units' which have the ability to be applied to any countable objects so long as they dl belong to the same defined class of things. This type of unit is useful when we count transitively, as we can apply it to any object of any size. The other kind of unit, the philosophical unit, has a further restriction placed upon it, namely that all units of this type must be of the same defined size. This, according to Plato, allows them to be used for intransitive counting - counting which 25 involves examining each individual to be counted in series (one, two, three, etc.) - and, therefore, as the basis of number. Regardless of which type of unit we consider, we are left with the idea that there are some basic properties which must apply in ail cases. First, any unit is by definition indivisible. This does not mean that it is impossible to divide a unit.

Clearly, if we have any sort of quantity it can be divided at least in thought. Thus, if we define a common unit as 'one potato', then we can find the number of these units in a bag of potatoes. There is nothing, however, which prevents us from cutting any given potato in half. All this would do would be to preclude our counting these halves as units. A similar argument seems applicable to the case of philosophical units. The second common is quantity. Thus, any unit, common or philosophical, must constitute a quantity of some son, even if it is an infinitesimal one. Finally, units cannot differ from one another. While this is easily seen to apply to philosophical units, since they are defined as identical, an explanation is in order for common units. Consider the unit of the potato again. Clearly, potatoes seldom, if ever, come in exactly the same size and shape. If we require that the unit be identical in every way, then it is going to be very difficult to find more than one of any given unit. However, all potatoes are alike in one very important way. They are all individuals of the same species. Thus, when common units are considered, we can ignore certain differences in individuals provided they all conform to some measurable standard, in this case, the standard of being a potato. Problems with the Traditional Interpretation of Points

1. The PIace of Points

With this broader knowledge of the properties of a unit, we can end the digression and return to the idea of a unit being a point without position, or without place, and see that it is problematic if not an outright contradiction. Leaving aside for a moment the problem raised earlier about points possibly having size, consider that we have already seen that all that a point possesses that a unit lacks is position. If we assume that points do not have size, though we have seen and will continue to see that this is a problem with which Aristotle grapples, then what are we to say is left when we remove position from points? Surely, they cannot even be units under the conception just given, as something must first exist in order to be considered a unit. If it does not exist, then it does not even meet the limiting requirement for being a unit, namely possessing quantity.

One solution we can consider at this stage involves looking carefully at the word

'&OETO<'. At first glance, one might assume that this means simply that a unit is a point without a set position. Thus, if we are looking at the world as a coordinate system, we might say that point A has position (x, y, 2). This would mean that every point would be assigned some specific coordinate along the spatial continuum. But this interpretation is fraught with difficulties. One is that this would seem to imply that Aristotle had a view of place as absolute; otherwise we could not identify a point with a single set of coordinates. However,

Aristotle's view of space is quite clearly a relative view. Thus, no set of coordinates is privileged in any way. Put another way, we could identify any point by assigning it a set of coordinates. Using this single point as a reference, we could then assign coordinates to all points based on the initial assignation. However, since space is relative, the same initial set of coordinates could just as easily be assigned to any other point in any other place and the corresponding relative assignations could be made from every possible initial state of affairs.

Thus, assuming we have an infinite number of points, and given that we seem to have an infinite number in a simple line this seems very likely, every point could potentially hold every set of coordinates. This means that any coordinates we might assign would be completely arbitrary. This is perhaps not as great a difficulty as it at first seems, though, for we might say that the actual coordinates are unimportant, being only a bookkeeping device used for our convenience.

A greater problem arises from the idea of points in . In a passage in De Anima

Aristotle tells us that

Further, since they say a moving line generates a surface and a moving point a line.. .36

Although this passage is polemic in nature, the idea of points being in motion is not unique to the De Anima. In the Physica, Aristotle mentions the idea of points in motion when he is describing exactly what motion is.

A diff~culty,however, may be raised as to whether a motion is specifically one when the same thing changes fiom the same to the same, e.g. when one point changes again and again from a particular place to a particular place."

Thus, we see that Aristotle did accept the idea of points being able to have motion. This would imply that even if we attempt a solution by saying that points can be identified by their relative positions in a given space, we would again face a difficulty when we introduced any geometrical object greater than a point. By doing this, we would have to allow for moving points. However, if a point moves, then it can no longer keep the same position relative to all other points. As soon as a point moved it would no longer be a point as it would have no position by which it could be identified. We might be able to trace such a

'rogue point' and identify it by a given position at a given time, but not only is this very complex but it would also seem to allow this point to assume the of any other point.

Consider that every location in space would be defined by a set of coordinates. Take, for example, a point at coordinate (0,0,1) which lies directly in the path of our moving point.

This would mean that at the time the point moved through (0,0, I), two points would take on the same coordinates. To make matters worse, assume that point (0,0,1)itself moves in order to create a line. What is left in its place?

This problem of a point having specific place is made more acute by the fact that in the Physica Aristotle writes

There is no necessity that the place should grow with the body in it, nor that a point should have place; nor that two bodies should be in the same place; nor that place should be a corporeal interval (for what is between the boundaries of the place is any body which may chance to be there, not an interval in body)."

This means that Aristotle is here saying that we do not need to have a point identified with place. This should lead us to the obvious question whether we, in fact, have defined '88~sog' 29

correctly. At first glance, to be sure, it would seem clear that we have. ' ' i48crog' is derived

from the verb 'ri0qp~'which means among other things 'place' or 'set': 'C&ro<' simply

means 'without position or place'. But 'without position' is in fact a phrase which is open to

interpretation.

We have seen that serious difficulties lie in the way of interpreting '&0eroq1to mean

'without defined position in a given space'. However, there is another possibility. ' ' METOG' could mean 'without position' in the sense that a unit cannot have position at alI. The distinction is less like the one between 'moral' and 'immoral' than the one between

'mordimmoral' and 'amoral'. By this I mean to suggest that since the idea of a unit with position leads us to this difficulty, perhaps it would be better to translate 'Zktoq' as 'not involving the concept of position'. If we interpret Aristotle in this way, then we can allow points to have position and to move without having to worry about assigning exact positions in a relative space. Thus, a unit may have position, but this is completely irrelevant to it being a unit. A single horse, taken as a unit, is one unit whether in a pasture or in a barn.

What is important is that it fits the kind established for a specific type of unit. On the other hand. a point must not only be a certain specific thing, such that it might be counted as a unit when we ignore its position, but it is important when we look at the point qua point that it does have position, though the exact position is unimportant. This understanding of '~~ETo<' seems to solve the two problems about the place of points.

2. The Size of Points

Having established that one property of points is position or place, we can proceed to examine the other properties points have. The traditional way of looking at the corpus asserts that Aristotle claims that points are the same sort of things that we today think of as points. That is, points are sizeless, massless entities which have only the property of position: they are geometrical entities, or at least proto-geometrical entities, which have no dimensions. However, while Aristotle does sometimes seem to discuss points as though they are entities of this sort. there are places in the corpus where we find properties attributed to points which do not fit this picture. For example, I have already alluded to the question of points having size. What will follow, then, is a further look at problem of this sort. I will examine those properties which Aristotle attributes to points and which are problematic for the traditional theory.

Ideally, the place to start such an examination would be with any explicit definitions given by Aristotle of what points are. In fact, he does give such definitions in the

Metaphysica. He writes there

That which is indivisible in quantity and qua quantity is called a unit if it is not divisible in any dimension and is without position, a point if it is not divisibIe in any dimension and has position, a line if it is divisible in one dimension, a plane if in two, a body if divisible in quantity in all - i.e. in three - dimensions. And, reversing the order, that which is divisible in two dimensions is a plane, that which is divisible in one a line, that which is in no way divisible in quantity is a point or a unit, - that which has not position a unit, that which has position a point.3g

In to its seemingly omnipresent definition as a unit with position, we have here a 3 1 more complex definition of a point. This, as noted by Evans, is a soa of definition by negation. We have a body divisible in 3 dimensions, a plane in 2, a line in 1, and that which is not divisible is a point or a unit." As it happens, we are in the unfortunate position of having little else which tells us explicitly what points are. The ey!icit definitions do not allow us to move very far. A different approach must be taken. This will involve looking less at how Arisotle defines points and more at the properties of points mentioned by

Aristotle. We have not settled the question about the size of points; we shall hope to settle it retrospectively after working on a further problem.

3. Points in Contact

Perhaps, given our overall concern with tangency and touching, the most important property of points on which we might dwell deals with the ability of points to be in contact both with one another and with other things. Both Physica E and Metaphysica K tell us that unlike units, which can only succeed one another, points can be in contact, though between any two points there is always something else, namely a line? Exactly how we are to interpret this is not clear. The problem is essentially that we do not have a definition of contact which seems to apply to points. Consider the meaning of term 'contact' as defined by Aristotle. We are told that

dxreo0a~62 6v[kcyo (s.c.)] zh d~pa6pa.

Things are said to be in contact when their extremities are together."

This means that for two points to touch, we must first find their extremities, and then we must have them close enough for their extremities to touch. But according to Aristotle's 32

definitions, both of these would seem to be impossible. In the first place, we are often told that points have no size. They are indivisible, and anything with size must be divisible.

Thus, since points have no size, they cannot have extremities with which to be in contact.

However, we did see that when we relate points to units there might be some size associated wi3 them. While this is not how we think of points, let's assume for the moment that

Aristotle would allow points to have some infinitesimal size, though not to the extent that they could be thought of as atoms. Recall that atoms, while actually indivisible, are potentially divisible into parts. For example, we can identify a right part and a left part of an . A point, however, is defined as being partless. Thus, if a point had size, it would have to have extremities, and it could then be said to have parts. Perhaps we can say that rather than having extremities as a sized object might have, a point would have extremities that were identical with each other and indeed with the whole point. In this way we might say that a point has infinitesimal size and is indivisible, but has extremities. Thus, when we had more than one point, we would have extremities which could be brought together to allow contact according to this definition. I must admit, though, that while we might accept such a solution, it seems entirely too farfetched to believe that Aristotle would have accepted it. To say that an entity has size, no matter how small, and yet that it cannot be divided even in thought seems to be the same soa of attitude which we see argued against, almost to the point of ridicule, by Aristotle (or at least by his students) in such works as De Lineis. Our only way out of this tangle would be to say that, despite what we have seen, points cannot in any way have size. Assume, though, that even without size we can somehow find extremities in points so that they could fulfil the conditions required for contact. The difficulty would be that of bringing these points close enough to touch. It seems, of course, that this is not possible. Since there is always something between any two points, we can always find a third point between our two chosen points. Thus, in order for two points to touch, they would have to be in the same place or else have other points between them. We are told of the ability of two points to be located in the same place in Physica A.

But this is not obvious as it is with the point, which is fixed. It divides potentially, and in so far as it is dividing the 'now' is always different, but in so far as it connects it is always the same, as it is with mathematical lines. For the intellect it is not always one and the same point, since it is other and other when one divides the line; but in so far as it is one, it is the same in every respect.43

Aquinas gives an interpretation of this passage.

He explains this by means of comparison with rnathematicd lines in which it is more manifest. In mathematical lines a point which is designated in the middle of a line is not always understood as the same. For insofar as lines are actually divided, they are contiguous. And those things are contiguous whose ends are together. But insofar as a point makes the parts of a line continuous, it is one and the same. For things are continuous whose end is the same?

We are told here that when a point joins two lines such that they become one line, then the resulting line is continuous. However, when we divide a line into two, we have two distinct points at the division. This makes the two lines contiguous rather than continuous.

Essentially, any time we have a situation where we have two points in relation to one another, they are either the same point or we have two points which are separated by some gap, making them unable to touch. This means that there is always some impediment to 34 points being in contact unless Aristotle is only speaking of whole-with-whole contact, which is, as we have seen, possible.

The idea of whole-with-whole contact is not an unproblematic notion in itself.

Essentially, this type of contact amounts to two bodies, be they geometrical or physical, coinciding. Thus, when we have two points in whole-with-whole contact, as we have supposed, we effectively have only one point. There are two difficulties with this idea which must be considered. The first is that it seems that by allowing two objects, even geometrical objects, to occupy the same position we would be violating one of Aristotle's prohibitions, namely that we cannot have two bodies in the same place. This applies even when we consider one object as a physical entity and one as a geometrical entity. Aristotle is quite clear in Metaphysics M that mathematicai objects cannot be in sensible things because it is impossible for two solids to occupy the same spacem4'It would seem that the best that we could hope for would be for the two objects to be contiguous. The other difficulty is the very problem which started this exploration in the first place. Suppose we have two points which are in whole-with-whole contact, each of which is on a different bronze sphere. When the spheres are in contact are the points in one sphere, in the other, in both, or in neither? In order to solve these problems we have two options. We can either say that two objects can occupy the same space, an option of which it is clear that Aristotle would not approve, or that points cannot even be in whole-with-whole contact, which explicitly contradicts Aristotle.

Or - the unacceptable possibility hanging over this whole enquiry - perhaps contact is not possible at all! 35

A Different Interpretation: Ideal Versus Red

Despite the problems we have seen associated with the view of points as sizeless, massless which possess position as their only attribute, this view is the one most commonly held by commentators on Aristotle. Evans writes

However, Aristotle is clear that a point is in no sense to be identified with a physical body, since a point has neither dimensions nor weight. Rather a point is indistinguishable from the place which it ~ccupies.'~

Further Ross writes

Aristotle treats these definitions of povti~and atlypj as implying the view that a unit actually becomes a point merely by acquiring position?

In other words, he affirms the traditiond line.

What I would like to do now is consider another possibility which might open the way to a solution to the problems raised earlier. As I mentioned at the start of the chapter, when we examine modern commentators on Aristotle, we find that they seem to say that

Aristotle simply takes the position that we would, that is, that the understanding of mathematics and geometry that we have is basically identical to Aristotle's. Thus, when we read his position on points, we would simply have to understand that he is talking about sizeless, dimensionless geometrical entities. I would like rather to suggest that Aristotle is clearly not so secure in his geometrical conceptions as we are; he is grappling with the idea of points, but his understanding of them has not settled into clarity. It is true that he seems to want to take the position that they are sizeless, dimensionless 'particles', but he also does not seem to be able to let go of the notion that there is something more to them. It is this other side, this 'something more' that I would like to consider.

Perhaps the most striking entry into Aristotle's other view comes in De Anima where he writes

It must all be the same whether we speak of units or corpuscles; for if the spherical atoms of Democritus became points, nothing being retained but their being a quantum, there must remain in each a moving and a moved part, just as there is in what is continuous; what happens has nothing to do with the size of the atoms, it depends solely upon their being a quantum.48

In this passage Aristotle is considering the movement of the . What I would like to concentrate on, however, is his astonishing remark that atoms that only possess the property of being quanta are nothing but points. Unless we regard this as a slip of the pen, and I think we should not, then this is a staggering blow to the traditional theory we have just seen. It is more akin to that which was identified as problematic during the examination of units.

Specifically, what we have here seems to be confirmation that Aristotle did in fact view points as having quantity. What I propose is to take the previous theory, the traditional Line which says that points have no size and no dimension, and consider it the ideal part of

Aristotle's thought. It is perhaps the theory to which he aspired and toward which he was working. However, what we are looking at here is what Aristotle was actually able to do.

Clearly, if we are to regard this passage from the De Anima as something Aristotle, rightly or wrongly, believed, then I believe that a fair conclusion might be that Aristotle was not as comfortable with the idea of a sizeless point as we would think. Adding to the strength of this conclusion is the fact that although he refers repeatedly to points as being without magnitude, he also on occasion makes mention of them as being more than nothing. 37

He differentiates in De Generatione et Corruptione between the possibilities of being nothing and being a point when he wrestles with the problems associated with division ad infiniturn. There he says that if such a division were to occur, we would either be left with a series of points, or with nothing.49 Even though this passage is one in which he is merely exploring the possibilities of how the objects of geometry exist, he is doing his best to exhaust those possibilities. By allowing nothing to be differentiated from points in such an enumeration, we see that he considers a point to be in some sense more than nothing.

Let us return, though, to the selection from De Anima. Recall that Aristotle mentions the possibility of removing from atoms all qualities necessary to make them points. Here he says that nothing would be retained except their being "quanta", .r;b ma6v in Greek.

Aristotle defines rb xoodv in Metaphysics A by saying

We call a quantity that which is divisible into two or more constituent parts of which each is by nature a one and a 'this'.50

However, De Lineis Insecabilibus is devoted to showing that there cannot be partless quanta, and especially that there cannot be indivisible lines. Even though De Lineis is generally taken to be a pseudonymous text, most likely written by one of Aristotle's students, when taken in conjunction with the above text from Aristotle it gives us reason to believe that it reflects

Aristotle's view - or at least a view which he held at one time. We are left, then, with a puzzle. If atoms which have every property removed except the property of being quanta are, by Aristotle's reasoning, points, then how can points be indivisible? This reminds us of the problem we dismissed earlier when we looked at contact between points. Recall that one of the difficulties there was that if we assumed that a point could have some size based upon the idea that any unit has size and, since a point is a unit with position, a point must also have size. The only possible answer to this difficulty is that in Aristotle's mind, though perhaps he would be 10th to admit it, there was still the vestige of the idea that points were indeed somehow sized entities.

Next, consider that we are told in De Caelo that points do not possess weight because they are indivisible. However, since we now have a situation in which points have magnitude, then we also have a situation where points can have weight. Consider the following passage.

For suppose that a body of four points possesses weight. A body composed of more than four points will be superior in weight to it, a thing which has weight. But what makes something heavier than a heavy thing must be heavy, just as what makes something whiter than a white thing must be white. Here the difference which makes the superior weight heavier is the single point which remains when the common number, four, is subtracted. A single point, therefore, has weight?

While it should be mentioned that Aristotle presents this as a reductio, in showing how absurd it would be for any indivisible object, for which he uses the point as a paradigm, to have certain properties, under this second theory it takes on a new meaning. Since we have points which possess size, we have yet another property for points which we were told we did not have. In the 'ideal' theory we were concerned with massless points. Suddenly, we are told that these points, between any two of which we can find a third, have weight! Taken to the extreme, we might suppose that a body can be as heavy as we like. All we need do is take the body made up of four points which Aristotle mentions and look for a point between two of them. We must be able to find one according to the definitions we have been given. Thus, we can find as many points as we need in order to add enough weight to satisfy us. As things stand thus far, we have two distinct ways in which we see Aristotle looking at points. First, we have the ideal theory, which more or less corresponds to our own conception. As well, we have the other theory, which seems akin to a form of atomism, whereby points possess the qualities of size and weight.

A further problem associated with points arises with respect to the idea that points, either qua geometrical objects or qua real entities, exist in every physical object. At first, we might think that this is a problem only for the reason that we have already seen, namely that we cannot have two objects existing In the same place at the same time. However, the problem is deeper still. In De Generatione et Corruptione, Aristotle writes

For since no point is contiguous to another point, magnitudes are divisible through and through in one sense, and yet not in another. When, however, it is admitted that a magnitude is divisible through and through, it is thought that there is a point not only anywhere, but also everywhere, in it; hence it follows that the magnitude must be divided away into nothing. For there is a point everywhere within it, so that it consists either of contact or of points. But it is only in one sense that the magnitude is divisible through and through, viz. in so far as there is one point anywhere within it and all its points are everywhere within it if you take them singly. But there are not more points than one anywhere within it, for the points are not consecuti-re; hence it is not divisible through and through. For if it were, then, if it be divisible at its centre, it will be divisible also at a contiguous point. But it is not so divisible; for position is not contiguous to position, nor point to point (i.e. division or comp~sition).~~

If the world is in fact so constituted, then we have the following situation. Every magnitude has points at every location within it. However, every magnitude is not divisible at every location within itself because this would imply, as we see in this passage, that we could divide any object at a point and at a contiguous point. We cannot do this, as this would mean that contiguous points could be in contact. However, we have already seen that this can only happen if they are in whole-with-whole contact. Thus, we are put in the situation of having to say that we cannot divide an object at any point we desire. If points are found at every location in every object and actually behave like this, then the world as continuous could not be possible in actuality. If it were, then it would be possible to divide a body anywhere and at a contiguous point. This having been ruled out by Aristotle means that once an initial division has been made, there must be locations where one cannot make a division. Once again, we have a situation in Aristotle's continuity theory that sounds suspiciously like atomism!

A way out of this difficulty, one which would allow us to save as much continuity as is possible in this theory, might . be found by referring to the fust few chapters of Metaphysics M, where Aristotle discusses the manner in which geometrical objects exist.

Here he makes mention of the fact that points, lines, and planes do not actually exist as part of any given object? If they did, then we would have more than one object in one place at 41 one time, an idea which we know that Aristotle considered absurd. Ross interprets Aristotle as saying that the mathematical objects exist, but in a qualified sense." Thus, when mathematicians analyze them, they are analyzing attributes held by objects, where those attributes are neither sensible nor non-sensible. This is because the objects of geometry exist merely potentially, and not actually. Since on this line of thought points exist only potentially, we might argue that points exist potentially at any location in an object, but that once a division is made certain potentialities are actualized. Thus, there are points at all locations in an object. When we select one, we must say that that particular point has been actualized. This means that no point can exist in a contiguous place. However, there is also nothing to preclude our initial selection of a point at the contiguous place. Were we to do so, we would then have the point actualized at this location, and any place contiguous to it could not contain an actual point.

If any of this is to make sense, then we must consider the second non-idealized theory as something real in Aristotle's mind. We must be able to say that there is some infinitesimally small region which surrounds an actual point in which there can be no other points. Sambursky, in comparing Aristotle's conception of place to the General Theory of

Relativity, claims

..a physical point is simply a singularity in the "metric field" which surrounds it."

This would imply that a physical point is, in some respect, a discontinuity in space. To say that a point could be merely a discontinuity avoids the problem but does not solve it. Recall that Aristotle's understanding of the infinite was such that it could only exist potentially. To have a point which could not be reached is absurd and forms at least part of the basis for his 42 rejections in the Physica of Zeno's Dicholomy and Achilles paradoxes. Thus, the idea of a discontinuity is not a possibility in Aristotle's mind. Now, if we are to say that we must surround the point with a region so that no other point can be contiguous to it, we must be talking about sized particfes. This must be so, even if the size of these particles is itself infinitesimal. This is because Aristotle defines contiguity in the following manner.

d~6uyevov6L 6 Bv k5fjq Bv &xrqrar.

That which, being successive, touches, is c~ntiguous.'~

If we had points with no size, then we would be able to make a division at any point regardless of location. This is so because Aristotle tells us that points can be in contact with one another (though only whole-with-whole contact) but cannot be in succession. This is because succession requires two objects which have no other object of the same kind between them, and between any two points one can always find another point." Thus, if points have no size, then at any location we choose there is a potential point just waiting to be actualized. We would not have to worry about making a division outside of a region defined so that no point can be contiguous to the point where the division was made. No point, save one that is in whole-with-whole contact, can be contiguous as there are an infinite number of points between any two selected points. However, if points have size but are indivisible, then such a thing is not always possible. To make a division in certain places would mean that we would be dividing a point. If there is a region where we cannot have any points, then what we have after we actualize a point is analogous to a jar filled with marbles. Our first choice determines where the points around it can or cannot be. We cannot cut through certain places without hitting a marble. However, there is nothing to say that we 43 could not have placed the marbles into the jar in such a way as to allow cutting at any given place, so long as we realize that by doing so, other options would then be closed. This is certainly true if we say that points have a definite size as opposed to the somewhat nebulous description of having infinitesimal size. When applied to Aristotle, this idea of an empty region reads, in a sense, almost like a form of proto-atomism, though one which allows for continuity, since the 'atoms', being mathematical entities, are merely potential.

One possible way to understand this strange view of real points versus ideal points might be to look at the locations where we find the problematic texts. In general, the texts about orrypj which support the view that Aristotle had an understanding of points which more or less coincides with our own view can be found for the most part in the Physica,

Metaphysica, and De Lineis. Those that do not are found, to a great extent, though not exclusively, in the earlier works and the biological works. Specifically, we find the idea that points are sized and, perhaps, visible in both Historia Animalium, and De Partibus

Animalium. This might give us cause to think that the proper translation for 'orryyfj' in the biological works would be something like 'dot' or 'speck' rather than 'point'. While this might work for some of the texts, such as the text about the speck of blood in an egg mentioned at the beginning of this chapter. most of the problematic texts are quite clearly about geometrical figures. For example, the text that concerns points as quanta comes from De

Anima. While we might try to say that Aristotle is thinking of dots here, it is quite clear from the context of the surrounding passages that he is discussing geometrical points. Further, the

Analytica Posteriora tells us that

...oiov povtg oiroicc &OETOG, onypij 6P, ocoia Bd5...... a unit is a positionless substance, and a point a substance having position ...58

Once again we see a refrain similar to the Pythagorean 'point without position' definition of unit, though from an earlier work than those which I have examined previously. This might give us the impression that Aristotle did, at least at some time is his early career, hold a view that was closer to atomism than the view he held later in his career. Further, if we accept the rough distinction I have made between the works, then we may have reason to believe that

Aristotle was not quite as comfortable with ideal points as we usually like to think.

Another way to look at this dichotomy in Aristotle's thought is to look at it purely as a problem of idealization. Rather than saying that Aristotle felt that points were entities which could hold contradictory qualities, such as having no size at the same time as being sized, we might say that all the difficulties we have in interpreting Aristotle boil down to a lack of refinement on Aristotle's part in thinking about geometry. What we see as a problem might be an overlapping of the qualities of ideal points with those of 'actual' points. That is, we might be able to make more sense of Aristotle's thought if we realized that at times he treats points as ideal objects of geometry, while at other times he treats them as though they are dots on a page. Ln fact, it has been suggested by Mueller that of the definitions we have of points and their properties, only some are to be understood as Iogicdly correct.

Aristotle's notion of how we grasp the idea of a point is not completely clear. Sometimes he says that the point is what has position and is indivisible; at other times he characterizes it as the limit or division of a line. The latter suggestion seems to fit better with the process by which we come to understand solids, planes, and lines. Perhaps the former should be thought of as the logically correct definition of a point rather than the description of our ordinary conception of it.''

This suggests that some of the definitions given are not to be taken as strict definitions. 45

Rather, at times we are given logical definitions which conform with points as we understand them, and at other times we are given less rigorous definitions which conform to a naive understanding of points. While I think that this is not a conscious decision on Aristotle's part

- I think given his methods he would likely try to be as rigorous as possible at all times - I do believe that this is a likely explanation. If we take this position, we still run into the same sons of problems we did earlier, but now we can explain our way out of them. Aristotle is trying to explain what points are and what properties they have. Though he might occasionally contradict himself, we can explain this by recalling that Aristotle is grappling with some sophisticated and does not seem to have a settled understanding of points.

This idea of thinking of Aristotle as being of two about such things as points to the extent of confusing and the ideal is not as outrageous as might first be considered.

What we must take into account is the fact that Aristotle was writing at a time when the mathematical and geometrical notions which we take for granted were still taking shape. It is not surprising to see him grappling with these notions and taking the occasional misstep.

This position can be understood if we look at Aristotle's position in the history of mathematics. Perhaps the most important, and generally overlooked, element is that while a great deal of work had been done up to the time of Aristotle, many of those on whose work much of modem geometry is based, such as Euclid and Archimedes, lived and worked after

Aristotle. In Aristotle's time the fundamental ideas of mathematics were still largely unsettled. Thus, we can start to understand his confusion when confronted with ideas of units and how they would differ from points. We can also begin to see where his attempts to be rigorous failed him, allowing him at times to see points as things with size, and confusing them with units. It is noteworthy that Mariarz and Greenwood have claimed, in their discussion of the Atomists, that

If we assume that Democritus tried to solve the Pythagorean difficulties concerning the incomrnensurables by means of his mathematical atomism, then a mathematical atom must be conceived as an indivisible. On the other hand, Democritus was too good a mathematician to be unaware of the snares of the indivisible lines. Yet according to Simplicius, he held to the indefinite divisibility of all lines. This view involves discrimination between mathematical and physical atoms, which was maintained later by , though Aristotle made no such distinction .60 (emphasis mine)

If this is correct, then Aristotle not only made the mistake of conhsing the real with the ideal

(as we might say), but he also confused the mathematical with the physical. We have already begun to see where these missteps were made. What remains is to understand why.

Ultimately, it would be better to give a more substantial argument than, "Aristotle was unsophisticated." I believe it can be shown that there is evidence for the position that

Aristotle was of two minds about the nature of points, and that this can be attributed to the differentiation made by Aristotle between physics and mathematics.

Aristotle defines the difference between physics and mathematics in the early passages of the ~h~sica.~~There he suggests that mathematics studies things such as lines, planes, and points, while physics studies bodies which incidentally have these objects in them. Thus, the two subjects are in the first instance the same in that they both study types of objects which have some connection to the objects of geometry, either by being such objects or by containing them in some way. What primarily differentiates them is the fact that the mathematician studies these objects in the abstract. That is to say that the mathematician does not examine these objects as inseparable parts of the physical world, but instead looks at them as ideal objects which are separate from the matter of the physical 47 world. This does not mean that these objects are without matter. Aristotle is clear that mathematical objects have matter, but of a different sort. To be precise, the matter of mathematical objects is called 'intelligible matter'. This idea will be examined in a later chapter.

The Advantages- of This Intermetation

At this stage, we are still unsure of what a point is in just the same way that we are unsure of exactly what a unit is. Clearly, a point in its purest form is a geometrical object.

It is difficult to know whether a unit can fall into the realm of geometry as well, or whether it is only protornathematical. Also, for Aristotle's predecessors a unit is the basis of numerical analysis. I believe that Aristotle held some regard for this position. However, I think that Aristotle, unlike his predecessors, went further in his analysis, conferring on the unit not just the property of being proto-numerical. but also that of being proto-geometrical.

Moreover, we have the additional problem of trying to distinguish between those properties which belong to points as ideal, mathematical objects as opposed to those which belong to points as physical 'marks'.

Returning to the questions we had earlier, how does this help us solve the problem of touching? At the beginning of this chapter I proposed to examine points and their properties with a purpose in mind. That purpose was to see how objects of a purely geometrical nature can come into contact. As we have seen, the problems involved are not simply physical. They are in part geometrical. Perhaps if we could understand points as being purely geometrical we might find a solution. However, points also seem to have a 48 physical aspect to them which I have attributed to Aristotle's insecure grasp of the idea of a pure geometrical point. Nevertheless, we can still find some of the answers we want.

Consider first the problem of tangency. From the start it has been maintained that when we look at touching from a purely mathematical point of view we do not run into any problems. When we consider points as purely mathematical objects, and this might conform to both theories which have been presented with little variation, this is true in a limited sense.

What I mean by this is that points can come into contact with each other. However, this contact is limited to whole-with-whole contact. Thus, when we have two objects of geometry, say a line and a circle, tangent to one another they are in contact to the extent that they actually share a single point. Since Aristotle does allow points to be in whole-with- whole contact this is still not a difficulty.

Our problems begin to reassert themselves when we recall that this was only the first in a chain of problems. If a point is shared between two objects, are the two objects still distinct, or are they now both part of a larger object? This applies as much for the objects of geometry as for the objects of the physical world. Recall that we have seen Aristotle assert that a single line can be divided into two lines such that they are contiguous when we have a situation where there are two points at the division where there was once one. This was attributed to the ability of points to be in whole-with-whole contact. If a line is a unity when connected by points in contact, wouldn't this make any two things connected in such a way a unity?

Further, we are still left with the problem of what happens to physical objects when they are in contact. We know that the objects of mathematics can be understood as abstractions. If this is so, and they are in contact, it would seem to mean that the method of contact is also an abstraction. Thus, we have two physical bodies sharing a place. This is not possible for Aristotle unless they are mixed in some way. The 'common' theory is of little or no help here, as these points behave either as the ideal points do, or more likely, they behave as other physical objects. Therefore, we must explore the idea of contact in greater depth, especially as it applies to physical situations.

Essentially, the problem at hand has taken on another dimension from that which it had at the start of this chapter. What began as an examination of what occurs when two points, the simplest of geometrical objects, come into contact has now become a question of how the behaviour of those points affects more complex objects. In other words, we are now concerned with what happens when two objects with some dimension come into contact.

While what has been accomplished in this chapter will be of importance to what follows, it gives us only part of the solution. What is required now is an examination of what happens at the limits of any object with dimension during contact. Thus, we need to explore the concept of boundaries.

1. It is interesting to note that Hume, in A Treatise of Human Nature Lii.3 (pp 38-39 in the Selby-Bigge edition), did not make this mistake when he asked what a simple and indivisible point was. However, this discussion does not directly concern Aristotle or his understanding of geometrical entities.

2. Euclid, Elements, in The Thirteen Books of EuclidS Elements, Volume 1, trans. Sir Thomas Heath (New York, Dover Publications, Inc., 1956) 155.

3. ibid. 153.

4. Constance Reid, A Long Way From Euclid (New York, Thomas Y. Crowell Company, 1963) 70. 5. RenC Descartes, Discourse on Method, Optics, Geometry, and , trans. Paul J. Olscarnp (Bobbs-Merrill Company, Inc., 1965) 177.

6. David M. Burton. The History of Mathematics; An Introduction (McGraw-Hill,1997) 143.

8. C. P. Jones, "Stigma:Tattooing and Branding in Graeco-Roman Antiquity,"The Journal of Roman Srudies LXXVTI (1987):139-155.

9. Metaphysica 10 16b24-27.

10. Physica 2O9a9- 13.

1 1. Metaphysica 1084b26.

12. Sir Thomas Heath, A History of Greek Mathematics, Volume I: From Thdes to Euclid (Oxford, Clarendon Press, 192 1) 69.

13. idem.

14. Melbourne G. Evans, The Physical Philosophy of Aristotle (Albuquerque, The University of New Mexico Press, 1964) 43. 15. ibid. 44.

16. Metaphysica 1052b20-24.

17. Julia Amas, Aristotle's Metaphysics: Books M and N (Oxford, Clarendon Press, 1976) 36.

18. Sir Thomas Heath, A Manual of Greek Mathematics (Oxford, Clarendon Press, 193 1) 38.

20. Heath, Manual 175.

21. Philebus S6d.

23. ibid. 8, note.

24. ibid. 37. 25. Heath, Manual 175.

26. Evans 44.

27. Metaphysica 1089b34-35.

28. W.D. Ross, Aristotle's Metaphysics (volume 2) (Oxford, Clarendon Press, 1953) 477- 478.

29. Analytica Posteriors 76a3 1 -3 6.

30. Annas 36.

31. Metaphysica 1053a24-30.

32. Annas 37.

33. Metaphysica 1016b21-24.

34. De Lineis Insecabilibus 97 1 a26-28.

35. De Lineis Insecabilibus 97 1 b20-26.

36. De Anirna 409a3-5.

37. Physica 227b 15-18.

38. Physica 2 12b23-27.

39. Metaphysica 10 16b24-3 1.

40. Evans 43.

41. Physica 227a.31, Metuphysica 1069a15.

42. Physica 226b23.

43. Physica 222a13- 17.

44. St. , Commentary on Aristotle's Physics, trans. Richard I. Blackwell, Richard I. Spath, and W. Edmund ThirIkeI (London, Routledge & Kegan Paul, 1963) 273.

45. Metaphysica 1076a37ff.

46. Evans 43.

47. W.D. Ross, Aristotle's Physics (Oxford, Clarendon Press, 1955) 628. 49. De Generatione et Corruptione 3 16a25-27.

50. Metaphysica 102Oa7-8.

52. De Generatione et Cormptione 3 l7a3- 1 1.

53. Metaphysica lO76a32- lO78b6.

54. Ross, Metaphysics (voLtme 2) 4 16.

55. S. Sarnbursky, The Physical World of the Greeks, trans. Merton Dagut (London, Routledge and Kegan Paul, 1963) 96.

56. Metaphysica 1O69a 1-2.

57. Metaphysica 1068b25ff.

58. Analytica Posteriors 87a36.

59. Ian Mueller, "Arkto tle on Geometrical Objects," Arch iv fur Geschichte der Ph ilosophie, 52 (1970):166.

60. Edward A. Marian and Thomas Greenwood, Greek Mathematical Philosophy (New York, Frederick Ungar Publishing Company, 1968) 68-69. Cha~ter3 - Boundaries

What is a Boundary? Starting from the definitions given by Aristotle, this section examines the concept of 'boundary' as something internal and as something external to a particular object. The former can be thought of as 'limit as form', while latter can be thought of as 'limit as surrounding object'. Though these seem to be coherent definitions, they lead to the difficulties involved with coincident boundaries mentioned in the first chapter.

The Problem of Coincident Boundaries An attempt is made to solve the problem by turning to definitions of 'place' and the relation of place to boundaries. This turns out to be a dead end and also creates problems for the notion of contiguity. Further, it ultimately entails allowing multiple bodies to exist in a single place. Thus, this line is abandoned.

The Problem of a Shared Boundarv A new approach is taken in order to solve the problem. This involves looking at definitions of 'touching' and seeing how this relates to limits. The difficulty of two boundaries becoming a single boundary again surfaces, leading to the question of how two continuous bodies differ from two contiguous bodies.

Aquinas' Solution - Touching Boundaries vs. No Boundaries The first attempt at making the distinction between continuity and contiguity is made by examining Aquinas' commentary on the Physics. This involves the distinction between contiguous objects, which have boundaries between them, and continuous objects, which do not. Unfortunately, this solution has limited applications at best. As well, it raises even more difficulties.

Another Solution - Boundaries in Contact vs. Mixed Boundaries Another solution is examined in order to solve our problems. This involves yet another examination of the concepts of continuity and contiguity. By means of this reexamination we are able to solve the part of the difficulty which involves differentiating between contiguous and continuous objects. However, some of the problems, especially those encountered with points, remain unsolved.

The Missing Piece of the Puzzle - Mixing The unsolved difficulties seem to rely upon the concept of mixing. 54

In the previous chapter the focus was on points and their relation to the problem of

touching. What was found was that when we consider points as purely geometrical objects

some of the problems of touching are solved. However. there are still some difficulties

which remain. The most severe of these seems to be differentiating between two objects that

are in contact at a point. This is a problem because, as we have seen, points as they are

ideally conceived can only be in whole-with-whole contact. Thus, when two geometrical

objects are in a state of tangency they must share a common point. Perhaps we might allow

for such an occurrence when we are speaking about geometry. This is not, however, so easy

to allow when we turn to the equivalent problem in the physical world. Suppose that we

have a wooden ruler touching a bronze sphere. This means that the ruler and the sphere

touch at a single point. If points can only be in whole-with-whole contact, then we have a

situation where a point of wood ada point of bronze share a single position. If we think

about this situation for a moment we find a host of problems. First, there is the obvious question of how two objects can share the same place. While we might be willing to allow such an occurrence mathematically, it seems to be precluded from occurring physically.

Next, assume that we allow this sharing of a point to take place. Do we still have two objects, or do we now have a single object? Put another way, are the ruler and the sphere still distinct, or have they become one? Consider that Aristotle tells us in the ~h~sica'that when two lines are joined by a point they become a single line. We have no reason as yet to

suggest that this should be any different when we consider physical object^.^

We can take this still further. We are able to consider the objects of mathematics in abstraction. That is to say that we can think of any element of geometry without having to 55 consider anythng else- This does not apply to physical things. Aristotle does not accept the idea of an actual void. Thus, there must be something at every possible location in the world.

For this reason the problem of tangency is only the tip of the iceberg of problems that will confront us. What really has to be considered is that every object has something else which touches it at every surface. Return to the ruler and the sphere. Not only does the ruler touch the bronze sphere at a point, but the sphere is also touched at every other point by the air which surrounds it. This adds to the problem of touching by involving the complexities associated with physicality. In particular, it us to consider the problem in relation to boundaries.

What is a Boundam?

In the last chapter we examined the point, the simplest of geometrical entities. The reason we consider points to be so simple is because they are dimensionless. They have, at least in their ideal versions, no size, weight, or shape. They are indivisible. When we consider geometrical entities of higher dimensions we encounter an additional wrinkle. In particular, we must deal with the concepts of limits. AristotIe tells us

od6~v6~yhp 61a~petoOxexepclaphou &v scCpcrq Lanv...

No determinate divisible thing has a single termination ...'

Since we need to deal with those objects which, unlike points, have parts and have more than one boundary, we must now turn our minds to an analysis of boundaries or limits.

Sharvy gives us his interpretation of what a limit is for Aristotle:

Specifically, a limit of something is the first point beyond which no part of that thing can be found, and the first point within which every part is contained. This "least upper bound" definition nicely leaves it open whether a limit of something is contained in it or not."

This is far ioo lax a way of looking at the concept. The implication is that it does not matter whether we define a limit as part of an object or as part of that which is not the object.

However, it will rapidly become clear that such a differentiation is of great importance to our task.

Aristotle gives an in depth definition of limit in the Metaphysics

We call a limit the last point of each thing, i.e. the first point beyond which it is not possible to find any part, and the first point within which every part is; it is applied to the form, whatever it may be, of a spatial magnitude or of a thing that has magnitude, and to the end of each thing (and of this nature is that towards which the movement and action are - not that from which they are, though sometimes it is both, that from which and that to which the movement is - and that for the sake of which), and to the substance of each thing, and the essence of each; for this is the limit of knowledge; and if of knowledge, of the thing also. Evidently, therefore, 'limit' has as many as 'beginning', and yet more; for the beginning is a limit, but not every limit is a beginnings

This tells us that a limit is one of at least four things. It is either the last point of a thing, the form of a magnitude, the final cause of a thing, or the limit of knowledge of a thing. Of these, only the first two concern us, as these are the only two which we can reasonably equate with the term 'boundary'. I would like to start with the second of these definitions - limit as the form of a

magnitude. The idea of boundaries being related to magnitudes must be clarified. The

reason for this is that it is all too easy to confuse the idea that an object has magnitude with

the idea that an object has place. We know that this is not necessarily the case, as clearly

points in their ideal form have place but no magnitude. However, when dealing with physical bodies these two concepts can easily become confused. Hence we can turn to the

Physica, where we see Aristotle wrestling with the problems of place and boundaries. He writes in Physica A that

The shape is supposed to be place because it surrounds, for the extremities of what contains and what is contained are coincident. Both the shape and the place, it is true, are boundaries. But not the same thing: the form is the boundary of the thing, the place is the boundary of the body which contains it.6

This suggests that boundaries can be defined in two related, yet different, ways. The first of these defines boundary as the form of an object. Essentially, this defines the concept of a boundary in what we might say is an internal fashion. That is to say that this definition of boundary depends solely upon the bounded object without regard to anything else. Thus, this type of boundary can exist in the absence of any other object. This means that this is the sort of boundary with which we must deal when we consider objects of geometry as objects of mathematics apart from all other things. The second definition relates boundary to place.

It tells us that place is a boundary which contains a body. In other words, this is an external 58

definition. It explains that a boundary depends upon the of some other object

besides the bounded object. This type of boundary must involve all objects of a physical

nature, as any physical object not bounded by another must be bounded by a void, and this

is not possible for Aristotle. As well, it may involve the objects of geometry, as it is

possible, for example, to create a situation similar to that of a chessboard where we have a

series of squares each forming the bomdary of other squares. These two ways of looking at boundaries - boundary as form and boundary as place - are, as Aristotle tells us in the above passage, not the same thing. However, they are related in that Aristotie later writes

Further, place is coincident with the thing, for boundaries are coincident with the bounded?

This suggests that although we have two different ways to define boundaries, they will ultimately yield the same thing. If we have an object with internal boundaries - that is, form - and with external boundaries - place - and place as a boundary is coincident with the bounded object, then we have a situation wherein the form and place coincide. This is not to say that they are the same thing. Clearly, form and place serve different functions and for this reason need to be differentiated. However, it would still seem initially that these two concepts are strongly related when we consider them in relation to the problem of touching. Therefore, a closer examination of both boundary as form and boundary as place may possibly help us to understand how it is that two objects can touch.

The first of these concepts - boundaries as form, making them an internal part of an object - probably has beginnings far earlier than Aristotle but which were adopted by him. In his discussion of Pythagorean geometry in his History of Greek Mathematics, Heath writes

A surface they [the Pythagoreans] called xpod, 'colour'; this was their way of describing the superficial appearance, the idea being, as Aristotle says, that the colour is either in the limiting surface (xbpaq) or is the xbpcr~,so that the meaning intended to be conveyed is precisely that intended by Euclid's definition that 'the limit of a solid is a surface?

This suggests that some aspect of Aristotle's thought about boundaries, at least as they relate to colour, can be traced back to the Pythagoreans. The passage in Aristotle to which Heath refers comes from De Sensu.

But it is manifest that, when the transparent is in determinate bodies, its bounding extreme must be something real; and that colour being actualIy either at the limit, or being itself the limit, in bodies. (Hence it was that the Pythagoreans named the superficies of a body its hue.) For it is at the limit of the body, but it is not the limit of the body; but the same natural substance which is coloured outside must be thought to be so inside

This seems to confirm the idea that Aristotle believed that the boundary of an object is not merely place as defined by a surrounding body, but is something which is part of any given body. This does not mean that the idea of place construed as limit was altogether dismissed.

Quite the contrary, as we have seen. This simply expands upon the idea of limit as part of a body. In this particular case, we see that any body with colour, including transparency, has an extreme that is itself coloured. This is important as Aristotle writes earlier that light is a property of transparent bodies which do not have a determinate b~undary.'~This passage tells us two important things. The first, something that is mentioned explicitly in this passage, is that colour itself is not a boundary. Colour is merely a property which may be

held by a limit. The second thing that we Ieam from this passage is that when considered as

parts of bodies Limits are real things. They are the surfaces of those bodies. They are not concepts to be considered only in the abstract. Still, even with this knowledge, we cannot say that we know what a boundary is.

We also see Aristotle discussing the idea of limit as form or as part of a magnitude in his discussion of place in Physica A. There he says

Now if place is what primarily contains each body, it would be a limit, so that the place would be the form or shape of each body which [sic] the magnitude or the matter of the magnitude is defined; for this is the limit of each body. If, then, we look at the question in this way the place of a thing is its form. But, if we regard the place as the of the magnitude, it is matter. For this is different from the magnitude: it is what is contained and defined by the form, as by a bounding plane. Matter or the indeterminate is of this nature; for when the boundary and attributes of a sphere are taken away, nothing but the matter is left."

This passage gives us the same two ways to define a boundary (xipa<) for an object that we saw in the other passages from the Physica. The first tells us that limit is not part of a body.

It is what contains a body, thus limiting it from without. The second definition tells us that the boundary of an object is part of the object, just as the matter it is made of is part of it. This is the same subtle distinction which allows Aristotle to have two types of boundaries.

The former is a boundary as defined by a container. Thus, we might say that it defines the boundary of an object as that which lies between the limits of another container. This is recognizable as the definition of place as the adjacent boundary of a containing body." The latter is boundary as defined by the form of the body. It is from this that we abstract to analyze the geometry of a situation. What I mean by this is that when we examine an object not qua physical body but qua geometrical object, such as when we look at a box as a rectangular prism rather than as a physical box, we are somehow abstracting from the physical to leave only those properties which would belong to the relevant geometrical figure. The precise method how this is accomplished will be dealt with in a later chapter.

What is important here is that by doing such abstraction, we are able to assess a physical situation in terms of its geometry. What allows us to work with the geometrical analogues of the physical bodies is the idea that the boundaries of physical bodies are in some way part of the bodies. Ultimateiy, these two definitions allow us to conclude that the distinction between boundary as fom- and boundary as place is really the distinction between limit of the thing and limit of the containing body.I3 By creating this distinction, we can now make sense of Aristotle when he writes in the Metuphysica

But if we suppose lines or what comes after these (I mean the primary plane figures) to be principles, these at least are not separable substances, but sections and divisions - the former of surfaces, the latter of solids (while points are sections and divisions of lines); and further they are limits of these same things; and all these are in other things and none is separable.''

This suggests that the surfaces of a substance are as much a part of a body as is the matter of the body. This fits with the idea that the limits of any body can be equated with the form of the body, without which we would simply have undifferentiated matter.

One thing that should be noted is that a body does not need have to have an external boundary.

For the limit must be the limit of something, but not necessarily in relation to something else: that which has a limit does not necessarily have it in relation to something else (as when it is limited in relation to the unlimited which comes next to it), but being limited means the position of extremities, and when a thing has extremities it need not necessarily have them in relation to something else. Some things, therefore, may happen both to be Limited and to adjoin something else, while others may be limited, but not in relation to something else. '*

This suggests that although a body may have both internal and external boundaries, there is no necessity that a body be bounded externally. In fact, as is pointed out in De Sensu during a discussion of colour and transparency, there is no necessity for a body to have an internal boundary either. Lf this is the case, then we do not have to worry about objects of geometry considered as abstract objects of contemplation with no other bodies acting as boundaries for them. It is clear that they can be considered using only their forms as their boundaries.

It should also be noted that by saying that a bounday or surface is part of a body, we 63

are not necessarily saying that there are two objects - a body and a surface - which exist in

the same place at the same time. We have already seen this suggested in the last chapter

when we considered the idea of points being in contact. It was said then that the only way

they could be in contact would be to be in whole-with-whole contact. This amounted to two points sharing the same place. Part of what allowed us to say that this was possible was the idea that points are geometrical in nature. It was when we started to attribute to them the properties of physical bodies that difficulties with whole-with-whole contact began to appear.

In this instance, we have to consider at least two different cases. The first is that of the boundary as a geometrical object. Here we should have little difficulty. Just as we can say that there are points at every place in a body, we can say that there are limits to bodies as well. These can be thought merely to be that geometrical shape which is left after the appropriate abstraction. The second case deals with limit of the body as a physical surface.

At Ieast initially, this seems easy to deal with. We might say that this is a mistake of language. A surface is part of a body. There is no more difficulty in saying that we have a surface and a body in the same place than there is in saying that we have a hand and a person in the same place. The one is part of the other. This having been said, is should also be noted that the idea of surfaces is still not entirely devoid of problems. All that I have claimed here is that we do not strictly seem to have to deal with the obvious probIem of two bodies sharing one place.

A Merproof that limits are part of a solid body comes from Apostle's analysis of

Aristotle. He notes that Aristotle makes the claim that if we divide a body into two parts, we do not have a greater quantity than we did before the division. For example, imagine cutting a gold bar with a very precise cutting tool, such that no gold is removed. If we assume that the surface of the gold is something above and beyond the body of the gold, then with every cut we will get an infinitesimally larger quantity of gold. This is because with each cut we increase the surface area. If the surface is counted as part of the gold, though not part of the actual body, then with each increase in surface area we have an increase, albeit a small one, of stuff- If we are to avoid generation of matter in this way, then we must maintain that the surfaces of a body, and hence the limits of that body, must be part of the divided body and not something extemal.16

This idea that the limits are part of a body adds further evidence to the notion that

Aristode held that there was no separate realm of geometry, similar to that place in which

Plato's Forms might have existed, but rather that all geometrical quantities are contained in physical bodies." This is what allows Aristotle to write

There are some who, because the point is the limit and extreme of the line, the line of the plane, and the plane of the solid, think there must be real things of this sort. We must therefore examine this argument too, and see whether it is not remarkably we& For extremes are not substances, but rather all these things are mere lirnit~.'~

When he says this we can now understand that the extremes conceived as geometrical objects do not exist as substance, but are more closely associated with form. It is only when we abstract them from the physical and consider them as geometrical objects that they take on even the remotest semblance of substance. 65

Although the idea of the boundary of an object being merely the form of that body is appealing to a degree, it is not without problems. Consider again the problem of touching.

If we suppose that two objects are to be in contact, then we must say that their extremes touch. However, if the limits of an object are part of that object, then when we bring two things together so that they are in contact we have a problem. When they touch, their extremes, not having any width, must be in whole-with-whole contact. Thus, the objects will share a boundary, making them in some respect, perhaps even in dl respects, continuous.

It seems absurd to say that any two objects in contact are actually continuous. The problem we had with tangency of points has returned to haunt us. We must enquire into this problem.

Before proceeding, some clarification should be made regarding points and their relation to boundaries. In the previous chapter, I raised the possibility of there being a dual interpretation of points in Aristotle's mind. My claim was that he had a theory to which he aspired and one that was more vulgar. The former dealt with points as entities without dimension, much the way we think of points today. The latter dealt with points as sized entities, similar to dots or even to atoms. The spectre of this second theory now returns to haunt us. If we even begin to think of boundaries as being made of points with size, then we will rapidly find ourselves in trouble. Any object with size must have a boundary. But if points form the foundation of limits, then we have points being bound by points which in turn must be bound by points, and so on ad infiniturn. While Aristotle does not have a problem with infinity in certain forms - after alI, one of the properties of the continuum is infinite divisibility - I think that this idea of boundaries of boundaries would be problematic.

It would seem to involve some sort of paradox, as we could no longer identify any boundary 66 because it would have a limit as well and this limit of the limit would then be the boundary of the object in question, and so on. To avoid this Zeno-like problem, I think it necessary to realize that when we look at points with respect to boundaries, we must confine our thinking to the ideal theory since, presumably, even common points have ideal boundaries. Thus, by solving the problems with the ideal theory, we can then treat the points as they are thought of in the common theory as we would any other physical body.

The Problem of Coincident Boundaries

Until now, I have concentrated on the definition of boundary as the form of an object.

I would next like to leave this and turn to one of the other definitions of boundary. The one

I would especially like to look at is the external definition which maintains that boundary is tied to place. In Physica A. Aristotle uses examples such as water in jars to explain exactly what place is. He writes

The existence of place is held to be obvious from the fact of mutual replacement. Where water is now, there in turn, when the water has gone out as from a vessel, air is present; and at another time another body occupies this same place. The place is thought to be different from all the bodies which come to be in it and replace one another. What now contains air formerly contained water, so that clearly the place or space into which and out of which they passed was something different from both.''

This example is not intended to show that boundaries exist. What we have here is an 67 example intended to illustrate that place exists. The claim is that when we remove all the air from a container by putting water in the container, the water effectively replaces the air at all locations in the vessel. Although the concept of boundaries is not explicitly discussed in this passage, the example given is important for us.

The problem lying in this example emerges from a passage later in the Physica:

Hence the place of a thing is the innermost boundary of what contains it."

Ross notes that this definition is incompatible with the above passage. We have seen

Aristotle tell us that the place of a thing contains it and that the outer surface of any body coincides with its place. This, according to Ross, means that body and place have the same size.'' However, this is only possible if the innermost boundary of a container is exactly the same size as the contained. For this to be the case, though, the boundaries would have to be in exactly the same place, making them the same boundary. The reason here differs from the idea that boundaries are equated to the form of the body in that here we are saying that body and place must coincide. Place. as has been suggested, is part of an external definition of boundary, while form is part of an internal definition. Thus we have a similar problem with the external definition to one seen earlier with the internal definition. Put another way, this creates a difficulty if we are to have objects which are merely contiguous, as according to

Simplicius

For those things are continuous whose parts meet at some common 68

boundary.. .=

This suggests that the analysis of the situation, namely that two objects which meet at a single boundary are continuous rather than contiguous by virtue of their sharing a single boundary, is the correct one. Aquinas agrees with this assessment.

For when two things which touch have one and the same terminus, they are said to be continuous."

If these commentators are right, then it would seem to indicate that any two objects which touch, as container and contained do, would have to be continuous.

One way in which we might redeem the notion of contiguity, especially as it applies in the case of liquid in a container, can be seen in a suggestion from Aquinas. He writes

For we find that something is said of a thing because of a part. Thus one is called white because his surface is white, and a man is called wise because there is wisdom in the rational part. If, therefore, we take a jar hll of wine as a certain whole whose parts are the jar and the wine, neither of the parts, that is neither the jar nor the wine, will be in itself. But the whole, that is, the jar and the wine, will be in itself insofar as each is its part, both the wine which is in the jar and the jar in which the wine is. In this way, therefore, it happens that something is in itself."

The possible solution given here applies specifically to objects being contained in other objects. When we consider a liquid in a container, Aquinas tells us that we are really looking at different parts of a greater whole. The whole in this case is the combination of jar and liquid. To think of the liquid as something distinct from the container is. according to

Aquinas, equivalent to saying that a property is something distinct from that in which the property resides. A container that does not contain anything is effectively an ex-container.

Ackrill gives an example from the Metaphysicn that explains this.

Certain objects, though countable, are by their very nature dependent - hands, for example. There are hands only so far as there are bodies with hands. Although hands are separable from bodies in a way - as colours and shapes are not - a separated hand is only an ex-hand. What it is to be a hand can be explained only by reference to the hand's role and function in an body.26

What this tells us is that a container must be considered as a container only when it contains something. As it happens, it must always contain something, as otherwise there would be a void. Thus, the container is in this way inseparable from what it contains. This does not mean that the contained objects cannot change. The passage which started this discussion clearly tells us that water can come to take the place of air. All that Aquinas' interpretation allows us to say is that when we consider a container, we can in some way think of what it contains as part of the greater whole. In this way, we no longer have to concern ourselves with the problem of touching, as we are not looking at the system as a set of objects, but rather as a single object with a number of parts. Obviously, though, this is less a solution than it is a means of avoiding the real problem. In fact, this solution seems to contradict the idea that an object cannot be in itself both primarily and either per se or per accidens.

Aquinas writes

A thing is said to be in somethingper accidens when it is in it because of something else existing in it. For example, we say that a man is on the sea because he is on a ship which is on the sea. He is said to be on the sea primarily because this is not due to a part. Therefore, if it would happen that something is in itself primarily, although not per se but per accidens, it follows that it is in itself because something else is in it. And thus it follows that two bodies are in the same place, that is, the body which is in it and it itself which is in itself. Thus a jar will be in itself per accidens if the jar itself, whose nature it is to receive something, is in itself and also that which is received - the wine - is in it. Therefore, if it follows that the jar is in itself because the wine is in the jar, then both the wine and the jar will be in the jar. And thus two bodies would be in the same place. Therefore it is clear that it is impossible for a thing to be in itself primarily.27 70

This tells us that despite what has been suggested, we cannot actudly consider the contained to be part of the container. To do so would again force us to the absurdity of claiming that two things could occupy a single place. In this case, we would have both the wine and the jar contained inside the jar. Since this effectively mles out the possible resolution suggested, this attempt at avoiding the issue falls apart.

The Problem of a Shared Boundw

Consider for a moment that for us, whose rough theory of the physical world might be called high-school atomism, the prospect that two objects which touch are not actually in contact does not fill us with dread? We are quite comfortable with the notion that eve+ng is made up of atoms and void, and that various bits are kept apart by means of different forces. Thus, when I consider the computer sitting on my desk, I am quite comfortable with the idea that the matter which makes up both the computer and the desk is mostly empty space, and that the rest is kept in place by various attractive and repulsive forces. It is true that this is not the way we commonly think of touching, but given our world view we can accept it. Aristotle, as an adamant foe of atomism, does not have the luxury of conceding the existence of empty space. Therefore, we would think that when he considered two objects to be in contact, there could be nothing, not even empty space, between them.

Hence, we must ask the question: Can two objects in contact have coincident boundaries, and if not, then how can Aristotle understand two contiguous boundaries as having nothing between them?

We are faced with a dilemma. On one horn we have acase where the boundaries of two objects in contact become one. This is problematic because it gives way to the mixing of the container with the contained. On the other horn we have two boundaries that are not shared, but are merely contiguous. This seems to lead to a situation wherein we either have a void between two objects or the two objects do not actually touch as there is something between them. AristotIe examines both of these situations.

Perhaps a good place to start would be to examine what Aristotle means when he claims that objects touch. At the end of Physica I? Aristotle differentiates between touching and being limited, but this is a difficult passage to contend with as it seems to give us some disturbing facts.

There is a difference between touching and being limited. The former is relative to something and is the touching of something (for everything that touches touches something), and further is an attribute of some one of the things which are limited. On the other hand, what is limited is not limited in relation to anything. Again, contact is not possible between any two things taken at random.29

We are first given a subtle distinction between touching and being limited, whereby touching is relative as anything which touches must touch something else. We are also told that this differs from being limited as being limited does not involve anything but the limited object.

At frst glance this seems to tie in with the internaVexterna1 distinction made earlier.

Touching another object is akin to the definition of boundary as place. We have the boundary of an object defined in relation to what surrounds it. Likewise, the definition of limit is like the definition of boundary as the form of an object. It is internal since it does not involvc anything but the object itself. Despite these similarities, this passage is problematic

when considered with respect to the problem of liquid in a container. This is because

dthough we have an assertion here that a limit is internal, we must make this cohere with the

idea that a container and what it contains may have the same boundary, thus making the limit

of an object relative as well if the liquid can be considered separately from the container.

The more disturbing problem with this passage is the idea that contact is not necessarily possible between any two objects taken at random. This second problem seems to have been solved reasonably by commentators. Ross refers us to a passage from

Sirnplicius.

06 yhp [dnreta~(s.c.)] ~cilQovfi ypcrppfjq, KCY~TOIlranCpavra~ k~btapov.~~

Simplicius illustrates by saying that an utterance and a line cannot be in contact though both are limited. Contact can only exist between two material things or between two mathematical objects, and is thus clearly distinct from imitation.^'

We also find Philoponus expressing a similar sentiment.

For a line is not in contact with a surface, nor a surface with a body, nor is a line in contact with a sound, but line is in contact with line and surface with

What this would mean for us is that when we consider the objects that are the focus of the problem of touching, we are always dealing with objects which can touch. The only time two objects cannot ever be in contact is when they are of two different kinds. Thus, an utterance and a line cannot be in contact, nor can an utterance and any physical object. The only way that a sound and an object might be in contact is if sounds were also objects of some sort as some Stoics seem to have suggested? Thus, it might also be suggested that a line and a physical object cannot be in contact because they are of two different kinds, one being physical and the other being geometrical. This is a problem, but it is not as thorny as might at first be thought. This is because we have already dealt with this problem in some respect when we considered the question of what a boundary is. There the claim was made. though the specifics were put off until later, that a boundary can be identified with both a physical aspect and a geometrical aspect of a body. Thus, when we say that a line touches a body, we can interpret this as the line touching the body when we consider the body as an object of geometry. In this way we avoid the problem, since like touches like. Given the complexity of the other problem, namely that of coincident boundaries between a liquid and a container, I will take for granted that this is the correct solution and move on.

Earlier, I mentioned the possibility that the boundaries of two objects in contact become one. Aristotle does not seem to have overlooked this possibility either. In the

Metaphysics he writes

For if substance, not having existed before, now exists, or having existed before, afterwards does not exist, this change is thought to be accompanied by a process of becoming or perishing; but points and lines and surfaces cannot be in the process of becoming nor of perishing, though they at one time exist and at another do not. For when bodies come into contact or are separated, their boundaries instantaneously become one at one time - when they touch, and two at another time - when they are separated; so that when they have been put together one boundary does not exist but has perished, and when they have been separated the boundaries exist which before did not exist. For it cannot be said that the point (which is indivisible) was divided into twos3'

This passage suggests that Aristotie believes that two boundaries which come into contact with one another do, in fact, change so as to become a single boundary. However, this seems to contradict what he writes in the Physica. There, when considering the problem of two objects in contact, as wine is in contact with the vessel which holds it, he says that the vessel is different from that which it contains.36 This means that even when in contact, two objects do not share enough to become part of one another. This might lend further credibility to the idea that boundaries are not part of an object but are associated with place. This would allow us to say that the boundaries that are shared are not part of the bodies in question, but are rather separate from those bodies in some respect. The problem with this is that we have already established that the distinction between boundary as form and boundary as place is not so clear that we can distinguish between the two in any practical sense. Thus, when we say that the boundaries are shared, we have difficulty saying that the place is shared but the form is not. If we were to say this then the wine would share place with the jar, but that they each retain their own forms. This would mean that we would have some sort of rn&uzge of a liquid and a solid in a single place.

Suppose that this problem is not so terrible and that we could have this mixture of solid and liquid at a boundary. We would still have a serious problem in that we say that when a liquid is in a jar it is in contact with the jar. However, if this contact amounts to a sharing of boundaries, we may not have contact, but rather we might have continuity.

Consider that after Aristotle gives us the example of the wine in a jar, he writes

What surrounds, then, is not separate from the thing, but is in continuity with it, the thng is said to be in what surrounds it, not in the sense of place, but as a part in a whole. But when the thing is separate and in contact, it is primarily in the inner surface of the surrounding body, and this surface is neither a part of what is in it nor yet greater than its extension, but equal to it; for extremities of things which touch are c~incident.~'

This puts us in the difficult position of having to say that while the two objects in contact

share a single boundary, they are not continuous. This is made even more diffkult by

Aristotle's assertion in the Metaphysica that

The continuous is a species of the contiguous; two things are called continuous when the Limits of each, with which they touch and are kept together, become one and the same, so that plainly the continuous is found in the things out of which a unity naturally arises in virtue of their contact."

Essentially, we are told that if the boundaries of two objects in contact touch and become the same, then we have a continuous object. Unfortunately, this seems to contradict the previous passage where we were told that when two extremities are coincident they touch. There is no assertion there that they become continuous. The only immediately apparent difference is that the passage from the Metaphysica tells us that not only must the limits touch, but they 76 must also become 'one and the same'. It is diEcult to see how this might help, as we would think that since no two bodies can share one place, any two bodies which share a place must become one. Thus, we are left with yet another problem: we must determine how it is that contiguity differs from continuity if both continuity and contiguity require the sharing of boundaries.

Aauinas' Solution - Touching Boundaries vs. No Boundaries

Part of the solution might be found in the Metaphysics. There Aristotle writes

A thing is called continuous which has by its own nature one movement and cannot have any other; and the movement is one when it is indivisible, and indivisible in time.39

On this Ross comments

The continuous is that whose movement essentially is and must be one, i.e. indivisible in time. Contact does not constitute continuity."

If this is correct, then we have a claim here that continuity requires contact, but that contact does not necessarily mean that we have continuity. The implication is that although we might have two boundaries in contact, this does not make the two bodies in question continuous unless the boundaries, by virtue of their being together, constitute a single continuous boundary. The difficulty here is that boundaries are not always divisible in every dimension. Take, for example, the boundary of a line. We have already seen that this is a point, which, when taken as ideal, is not divisible in any dimension. This makes it rather hard to see how it is that when two lines share a boundary this boundary can be divided. Certainly lines can be divided, and a division can be made at a point, but a point itself cannot be divided. Likewise, since we have already seen the assertion that lines are the extremes of planes and planes are the limits of solids, we can easily extend this difficulty to higher dimensions. This would seem to mean that when we say that we have two boundaries together, namely in whole-with-whole contact, then what we really have is a single boundary.

How, then, if we are told that we have a single boundary between two objects in contact, are we to make sense of the idea that contact between boundaries does not imply that they are continuous?

Return for a moment to the passage from the Physica about the surface of a containing body being neither part of what it contains nor greater than its extension. Aquinas comments on this passage. He says that

[Aristotle] says, therefore, first that when the container is not divided from that which is contained, but is continuous with it, then the contained is not said to be in the container as in a place, but as a part is in a whole. For example, let us say that one part of air is contained by all air. He concludes this from the foregoing, for where there is a continuum there is no extremity in act, which he said above was required for place. But when the container is divided from and contiguous to that which is contained, then that which is contained is in place, existing in the extremity of the container primarily and per se. And the container, which is not part of the thing, is neither larger nor smaller in size, but equal. He explains how the container and that which is contained can be equal by the fact that the extremities of things which are contiguous exist together. Hence their extremities must be equal?

Although Aquinas is trying to clarify things here, this passage only raises more questions.

We are told that when we have a puff of air in a larger body of air, we in fact have a situation where we do not have a container and a contained. We have instead a part within a greater whole and, therefore, no boundary at all. Thus, what differentiates a continuum from two 78 objects in contact is the absence of any boundary in the former. However, when we have a body contained within another body, such as our wine in a jar, we have a division between the bounding and the bounded. This is enough to constitute contiguity but not continuity even if the extremities are in the same place. The most obvious question to be asked here is why, even if we were to accept this for similar fluids, such as air contained in air, should the same solution be applied to dissimilar objects? That is, suppose that we have a drop of wine which is mixed with a larger quantity of water. At some time during this mixing we are told that the form of the wine will dissolve and the wine itself will become part of the water." At what point does the boundary between wine and water cease to exist? If it is prior to the change, then the wine must be continuous with the water. If it is after the change, then the water must be bounded by water. If it is at the exact moment of change, then the boundary must instantaneously cease to exist. We know this is not problematic because of the discussion of change, becoming, and perishing in the Metaphysics which was mentioned earlier. There we find an assertion by Aristotle that a boundary can come into being instantly, and perish instantly as well. While this may work on the level of boundaries, it is problematic when we try to find out how it applies to a physical object. While we may be able to say that two drops of water become one bigger drop instantly when the boundaries between them perish, we cannot say that they become one drop instantly. This would imply a change which goes beyond the perishing of boundaries. It involves a change in size and shape. We know that this kind of change requires time. Do the drops stop being contiguous and become continuous instantaneously, or does it happen over time?

Barring the problem of dissimilar objects, it might seem that the above passage gives what we might consider an obvious solution to our problem. When we say that two boundaries touch, we say that there is nothing between them. Thus, the two boundaries which existed prior to contact become a single boundary. Despite this, neither bounded object becomes part of the other even though they both share a boundary. Unfortunately, this raises a problem with the idea that the boundaries can be together in virtue of their being contiguous. Aristotle has said that when in contact but not continuous, the boundary is not part of the second object, but it is not greater than this object either, and in this passage

Aquinas affirms this. However, if we look at the definition of contiguous given by Aristotle, we find a problem.

A thing is in succession when it is after the beginning in position or in form or in some other respect in which it is definitely so regarded, and when further there is nothing of the same kind as itself between it and that to which it is in succession ... A thing that is in succession and touches is contiguous. The continuous is a subdivision of the contiguous: things are called continuous when the touching limits of each become one and the same and are, as the word implies, contained in each other: continuity is impossible if these extremities are

This implies that only the extremities of continuous objects can be equal, while those of contiguous objects must be unequal in some way. Consider the case of wine in a jar. We have seen that the boundaries of the wine and the jar cannot be united, as then they would be continuous, an impossibility. Suppose, then, that they are merely contiguous. According 80 to the definition, this means that they are in succession, so there is nothing of the same sort between them. I assume that this means that there is no other boundary between the two in question. We can easily accept this. However, the second part of the definition, saying that the two objects touch, begs the question. We are trying to find out what it means for two objects to touch. There are three possibilities: the fust is that the boundaries are contiguous.

We have already dealt with this possibility. Recall that for two things to be contiguous, they must be in succession and must touch. The problem is that this leads us in a circle. We are attempting to discover what it means for two objects to touch. Therefore, we cannot say that if they are contiguous they touch without giving a circular definition. The second possibility is that there is something other than boundaries between the two boundaries, such that they remain in succession. The third possibility is that there is nothing at all between the boundaries, but that they fit together perfectly.

The second possibility does not seem to fit our understanding of touching, but is worthy of furrher examination. If we are to have some object between the boundaries which does not create more boundaries, then there are at least two possible things which can be between them. One object that exists without a boundary is the ubiquitous point. Thus, we can suppose that between any two boundaries we might have a point, or perhaps even many points since they take up no space. At first we might think that this would imply that the two objects were not touching in the normal sense of the word. Rather, they would both be touching a point. Upon closer examination we would find a more insidious problem returning to haunt us. Since contact with a point must be whole-with-whole contact, this would seem to mean that the two boundaries must also be in whole-with-whole contact with 8 1 each other. Thus, the boundaries must share the same place with the point, and can only do this if they are continuous. But this means that the possibility of succession reduces to that of contiguity with all of the associated problems.

I claimed in the previous paragraph that there were at least two things that existed without a boundary. As mentioned earlier, one of these is the ideal point. However, we might say chat a point is a limiting case as it has no dimension. This is part of the reason that the examination moved to more complex objects. We must at least consider the possibility that the situation would be different if we had an unbounded object which took up space.

Lest we think that such a thing does not exist, recall that I have already mentioned the case of light. While we are told in De Anima that light is the colour of what is tran~parent,~

Aristotle also tells us in De Sensu that

fi pEv ohroc @wtb~@6o~q Bv aopiorq rQ 66rcc@ccvsi 8asiv*

...light is a nature inhering in the transparent when the latter is without determinate boundary?

This allows us to make the claim that light is the property of a transparent medium, not of an obje~t."~This does not mean that all transparent bodies are lit. Aristotle is quite clear that when a transparent body is a determinate body it has a determinate limit, which must mean that, like the common transparent bodies of air and water, it is not necessarily lit. However, we are not concerned with limited bodies. Our concern here is with things that can exist without limits. Thus, light seems to qualify as something which has place and which is unbounded. This would tell us that between any two successive boundaries we could have light. Further proof that this might be correct comes from a passage in the Physica. A thing is moved continuously if it leaves no gap or only the smallest possible gap in the material - not in time (for a gap in the time does not prevent things moving continuously, while, on the other hand. there is nothing to prevent the highest note sounding immediately after the lowest) but in the material in which the motion takes place."

Although this passage is more directly related to motion, the idea seems applicable to the case at hand. The bizarre suggestion here, one that does not seem to mesh with our understanding of continuity at all, is that if an object has a gap of some sort in it, it can still be continuous. Since continuity is a species of contiguity, it stands to reason that if such a condition can hold in the former, it can probably hold in the latter. The problem with this is that it does not really seem right to say that two objects are touching when they are separated enough such that light can pass between them. Further, since a gap in an object would imply that there is a limit at the gap, we would again be left in a position of saying that two objects are touching when they are not actually in contact as we understand it. While it is true that if we accept Aristode's assertion that a continuous object can have a gap in it we might say that Aristotle simply had an unusual way, to our minds at least, of thinking of touching, I think it is probably better to forgo such a radical move and consider this passage something of an aberration unless we are forced to accept it. Therefore, it seems, that using succession to explain touching will not work.

This leaves only the third possibility, namely that boundaries fit together perfectly, leaving space for nothing in between them. If we are considering two straight surfaces, then 83

accepting the solution Aquinas gives us, namely that the extremities of the bounded and the

bounding objects are equal, we may not have a problem. Any two parallel lines on a plane

can have the same length. However, the problem given us by Aristotle does not deal with

iines on a plane. Aquinas' solution is a result of the problem of a container and the

contained, such as wine in a jar. Clearly, then, we must be dealing with more complex shapes than lines. For the sake of simplicity, suppose that the boundaries of the wine and the jar are both curved surfaces. This means that if the two are flush, then the inner surface must be shorter than the outer surface by some amount, even if that amount is of the smallest measure. The situation would be something like taking the skin off of an onion and putting the whole skin in its original shape next to the onion. To the eye there might not seem to be any difference in size, but we know that the skin must be ever so slightly larger than the onion. Thus, the boundaries, unless they are in exactly the same place, must be of slightly different sizes. This means that Aquinas cannot be correct in his statement that the extremities must be equal unless the boundaries do indeed become one.

Another Solution - Boundaries in Contact vs. Mixed Boundaries

One interpretation that might offer a solution can be found by reexamining the definitions of contiguity and continuity. Lang notes that Aristotle is clear about the fact that when things touch, their extremities occupy one place and nothing can intervene. However, when things are continuous Aristotle says that their extremities become "the same and one"

("... Adyo 6' dvcn auve~bsijtav tctljrb ydvqrar ~aiBv ...") and that continuity is impossible if their extremities are two. Thus, Lang notes, two lines which are connected and become continuous form a single line? Up to this point, I have claimed that the implication of two extremities, be they points, lines, or surfaces, occupying a single space is that they effectively become a single extremity. In fact, at times the boundaries are no longer even considered extremities after contact is made. Take the example of two lines that are connected and become a continuous line. In this case, the limits of the two original lines may stiU exist, but they are no longer considered limits. Since they must be in whole-with-whole contact - since the limits of lines, namely points, cannot be in any other type of contact - they are at best simply a single point which can be identified on the new line.

The reason that I have returned to the idea of extremities becoming one and the same is that although this seems initially to cause more problems for us, it may actually help us to find a solution. In the Physica Aristotle writes

Things are said to be together in place when they are in one primary place and to be apart when they are in different places. Things are said to be in contact when their extremities are togetherS4'

Commenting on this passage, Ross says

...two things are By cr if they are in one place which contains nothing but the two,i.e. where there is nothing between them."

As well, Urmson, in his translation of Simplicius on the Physica, says in a note that

This is meant to fall short of the extremities being one, which yields contin~ity.~'

When we put these passages together we start to see a possible way out of our difficulties.

We are told that contact allows for two things to be together but not to become one thing.

If Urmson's interpretation of Simplicius is to be followed, then we were correct in the earlier 85 assessment that two things becoming one and the same thing would give us a situation akin to continuity. The picture we can now paint is similar to one of the earlier options which was discarded, namely the third option with which Aquinas' solution presented us. This option was that we can have two limits together that do not allow anythmg between them. One important difference here is that we are making no assertion as to the relative sizes of the boundaries. All we are saying is that they are as close as they can possibly be without actually becoming one.

If this is the case. then we have two similar situations both suggested by Aristotle when he tells us in the Metaphysics that the continuous is a species of the contiguous.s2 The first, and stronger of the two, is that of continuity. In this case, two boundaries come into contact and mix to such a degree that they become one, or possibly disappear altogether. In the second case, that of contact, any mixing of boundaries which takes place is much weaker.

By weaker here I mean to say that unlike the case of continuity, there still exists a boundary between the two objects and the boundary itself is separable into the two original limits.

In order to confirm this possibility, we can tun back to Aquinas. When he discusses how it is that elements change into one another he says something that, if interpreted in a certain light, may help this theory.

Therefore, when air and water are two distinct things, they touch. But when one thing comes to be from either of them by one changing into the nature of the other, then a union or continuity comes to be.')

This seems to indicate that touching only applies when we have two types of matter which do not become Like one another. This would imply that any mixing of the two elements will not take place. Thus, when water does not become air we have only touching. However, if 86 one element changes into the other, or presumably, when both mix so as to form another compound, they take on some shared properties. When properties that were not always shared come to be shared, we have continuity.

We can make further sense of this when we consider a continuous figure such as a cube.

The form of a cube is its six bounding squares, and the matter is what these squares bound: a formless and continuous interval, which may be called "depth." This is the ultimate intelIigible matter of the cube; for the cube may also be considered as a composite of two half-cubes or of any number of such parts each of which possesses matter and (at least potentially) form."

If we consider this, what we find is that any division is potentially possible in a continuous object. It is only in the contiguous that the division has been actualized. It is for this reason that Apostle makes a point of saying that two squares which have sides which coincide externally are contiguous, but two semicircles are continuous because not only are their limits one, but they also only have potential existen~e.~'Thus, if we assume, as Aristotle does, that the process of division does not involve anything being added to or taken away from the divided solid, we have licence to say that the actualization of any division creates contact from continuity, and the separation of the parts creates succession. This only solves part of the problem, as we still do not know how two objects brought together could find themselves in the same place so as to become contiguous without violating the rules of physics.

On a less complex level, we aiso still have the problem associated with points coming into contact without giving us continuity. The solution I have presented here relies upon the idea that two objects can come into contact in such a way that there is not!ing between them, 87 and that they are only continuous when there is some sort of change sufficient to allow a sharing of properties. Once again, points are a problem as any contact between them must be whole-with-whole. We cannot have two points, at least not two ideal points, next to each other without anyhng between them, as we already know that between any two points there is a third point. This means that any time two points are in contact, there must be mixing of some kind. Extended to a higher dimension, we cannot have parallel lines in contact without mixing. Once again it seems that the geometrical aspects involved with touching are the downfall of the solution.

The Missing Piece of the Puzzle - Mixing

All hope is not lost. There are several concepts which have come up that may lead to a solution. Among these is the idea of mixing. I have suggested that the solution to the problem of touching may involve limits mixing or not mixing when they come together. The problem just raised was how to keep points which are in contact with one another from mixing. If this is possible then we may indeed have the correct solution here. Further, it may not even be required that points not mix. Since we are concerned with points as geometrical entities, there may be a way for them to touch without our having to disqualify the theory for physical objects. What is required is a more in depth look at the core of this possibility, namely the idea of mixing. It is this concept that will be examined in the following chapter.

2. Some of these problems were raised by Pierre Bayle in his article on Zeno of Elea in the Historical and Critical Dictionary. However, Bayle presents them in order to show how those who follow Zeno should argue against extension and motion, rather than to illustrate the difficulties which I have identified in Aristotle's thought. Thus, though he mentions Aristotle and his discussion of Zeno's paradoxes, Bayle's discussion focusses on the problems raised by Zeno rather than the potential solutions which might have been offered by Aristotle.

3. Physica 2 18a.22-23.

4. Richard Sharvy, "Aristotle on Mixtures." The Journal of Philosophy, WM (1983): 448.

5. Metaphysica 1O22a4- 13.

6. Physica 211b10-14.

7. Physica 2 12a29-30.

8. Sir Thomas Heath, A History of Greek Mathematics, Volume 1: From Thaies to Euclid (Oxford, Clarendon Press, 192 1) 166.

9. De Sensu 439a28-b1.

10. De Sensu 439a27.

1 1. Physica 2O9b 1- 1 1.

12. Max Jammer, Concepts of Space: The History of Theories of Space in Physics (Cambridge, Harvard University Press, 1954) 16.

13. Friedrich Solmsen, Aristotle 's System of the Physical World: A Comparison with His Predecessors (Ithaca, Comell University Press, 1960) 126.

14. Metaphysica lO6Ob 12-17.

15. De Meiito, Xenophante, et Gorgia 978b 10- 17.

16. Hippocrates George Apostle, Aristotie's Philosophy of Mathematics (Chicago, The University of Chicago Press, 1959) 102.

17. Jonathan Lear, Aristotle: The Desire to Understand (Cambridge, Cambridge University Press, 1988), 23 1-232.

18. Metaphysica 1090b5-9.

19. Physica 208bl-8.

20. Physica 2 12a20-2 1. 2 1. W.D.Ross, Aristotle's Physics (Oxford, Clarendon Press, 1955) 576.

22. Simplicius, In Aristotelis Physicorum: Libros Quattuor Priores, Commentaria, 748.3.

23. Simplicius, On Aristotle's Physics 4.1-5, 10-14, trans. J. 0. Urmson (Ithaca, Cornell University Press, 1992) 159.

24. St. Thomas Aquinas, Commentary on Aristotle 's Physics, trans. Richard J. Blackwell, Richard J. Spath, W. Edmund Thirlkel (London, Routledge & Kegan Paui, 1963) 3 14.

25. ibid 201.

26. J. L. Ackrill, Aristotle the Philosopher (Oxford, Clarendon Press, 1994) 121.

27. Aquinas 202-203.

28. It is important to note that this reference to modem atomic theory does not specifically refer to the view of atoms that was abandoned by scientists at the end of the 19" century. Atomic theory as held at that time did not involve dynamic forces or field phenomena, but, since atoms were thought to be absolutely hard, did involve repulsion in surface contact. The project of atomism was to get rid of forces which were allied with Aristotelian active and passive potencies. My intent with this reference is to make a claim about how the average person at the end of the 20b century with some understanding of the matter tends to understand how the world is composed.

29. Physica 208al1-14.

30. Simplicius, In Aristotelis Physicorum, 5 16.27.

3 1. Ross 562.

32. Philoponus, In Aristotelis Physicorum: Libros Tres Priores: Commentaria, ed. Hieronyrnous Vitelli. Vol. 16. Commentaria In Aristotelem Graeca. (Berlin: G. Reimer, 1887) 494.15-17.

33. Philoponus, On Aristotle's Physics 3, trans. M. J. Edwards (Ithaca, Cornell University Press, 1994) 155.

34. A. A. Long and D. N. Sedley, The Hellenistic Philosophers, Vol. I: Translations of the Principal Sources, With Philosopica1 Commentary (Cambridge, Cambridge University Press, 1995) 195ff.

36. Physica 2 10b27. 37. Physica 2 11a29-b34

38. Metaphysica 1069a.5-8.

39. Metaphysica 1016a5-6.

40. W.D. Ross, Aristotle's Metaphysics, volrirne 2 (Oxforci, Clarendon Press, 1953) 300.

4 1. Aquinas 206-207.

42. De Generatione Animalium 328a27.

43. Physica 226b34-227a13.

44. DeAnima418b12.

45. De Sensu 439a27-28.

46. John Thorp, "The Luminousness of the Quintessence," Phoenix 36 (1982): 122.

47. Physica 226b27-3 1.

48. Helen S. Lang, Aristotlek Physics and its Medieval Varieties (Albany, State University of New York Press, 1992) 5 1.

49. Physica 226b2 1-23.

50. Ross, Aristotle 's Physics 627.

5 1. J. 0. Umson, translator's note, On Aristotle S Physics 4.1-5, 10-14 by Simplicius, 66.

52. Metaphysica 1069a5.

53. Aquinas 222.

54. Apostle 106-107.

55. ibid. 1 18- 120. Chapter 4 - Geometrical Obiects and Intelligible Matter

Why the Subiec: of Mixing is Important to the Project A brief explanation is given to explain more clearly why mixing needs to be examined. This involves revisiting some of the earlier problems encountered. Also, a connection is established between touching in a physical context and touching in a geometrical context.

AristotIe's Views on Mixing An examination of what Aristotle had to say on the subject of mixing. Particular attention is given to the terms 'rptolq' and 'piE~q',with an eye to differentiating between their uses.

Difficulties with Aristotle's View There are four main problems with Aristotle's understanding of mixing. The most obvious is the problem of coincident boundaries, which caused great difficulty in the last chapter. Beyond this. there is the difficulty of the time during which mixing takes place, and the problem of new material being created at a boundary if we allow mixing to take place between two objects that touch. Finally, there is the need to be sure that any solution which involves mixing take into account both physical mixing and geometrical mixing.

Intelligible Matter - Solving the Problem of Geometrical Mixing; A digression is made so as to examine the concept of intelligible matter. Using this concept, along with a better understanding of how the objects of geometry are abstractions from the physical, we gain a clearer understanding of how geometrical objects exist. In this way, an explicit connection can be made between physical and geometrical objects.

The Difficuity of Making the Transition from Physical to Geometrical Obiects This section smooths out some of the rough edges involved with the concepts of intelligble matter and abstraction. This allows us to use these concepts to fully understand the relationship between physical and geometrical touching.

Solving the Other Difficulties This chapter has only solved the difficulty of making the connection between the physical and the geometrical. In order to solve the other problems we need to explore the concepts of actuality and potentiality. 92

Whv the Subiect of Mixing is Important to the Proiect

The major problem explored in the last chapter was that of finding out what happens to boundaries when they are in contact. Aristotle tells us that

And if there is continuity there is necessarily contact, but if there is contact, that done does not imply continuity; for the extremities of things may be together without necessarily being one; but they cannot be one without necessarily being together.'

This suggests that the idea that the boundaries of two bodies must in some way mix when they touch is wrong. If extremities can be in contact without being together, then we do not have to worry about mixing unless we have continuity. Ln fact, we are also told that

A thing is moved continuously if it leaves no gap or only the smallest possible gap in the material - not in the time ...but in the material in which the motion takes place.2

This might be taken to mean that we do not need mixing even in the case of continuous objects. It might be possible to regard the second passage as an aberration, since it seems clear that we require contact between objects in order to have continuity and having a gap between objects does not seem to fulfil this requirement on any of the terms we have seen

Aristotle give us. However, this still leaves us with the first passage. The problem with this passage is that it leaves contact open to the problems we have seen earlier. Suppose that there are two boundaries in contact. If they are not together, then they must be in different places. But, as we saw when we examined points, this would mean that something could 93 come between the two limits. To say that the boundaries are flush with one another does not

solve the difficulty, since we know that the only way that an object without dimension can

be in any state which approximates being against something else is to be in whole-with- whole contact. Boundaries, being for Aristotle either points, lines, or planes, are objects that have at the most two dimensions. Thus, they must be in whole-with-whole contact when they touch. The situation with which we are left is one where we must either say that the objects are together but do not mix, and I do not see how this is possible, or that the passage above is a mistake on the part of Aristotle and that when two objects come into contact they do come together and must in some way mix. This requires further explanation, as the concept of mixing is a complex one in Aristotle's thought, and thus is not a subject that should be used as a solution without an explanation. Consider, then, the solution presented as it relates to mixing.

One of the points made in the soIution was that when two boundaries come into contact, they may mix so that they become one boundary, or so that they cease to exist as boundaries altogether. Simply asserted in this way, this is inadequate. There are clearly too many problems to be simply ignored. The most obvious one, given the way I have stated the solution, is that we need to understand the difference between those objects that become continuous when they mix and those that only mix at the boundaries, thus remaining contiguous. Further, there is the problem of how boundaries can be divided. In other words, we must determine whether a mixture can be returned to the components of which it is made, and, if so, how this is done. If it is not possible to separate mixtures, then we cannot say that the solution presented is viable. To say so would be to allow for objects to come into 94

contact, mixing at the boundaries, but they would never be able to leave contact without

leaving a part of themselves behind in the mixture. Finally, we must consider the problem

of mixing as it applies to the abstract as well. As will be seen, when Aristotle considers

mixing, he does so for physical objects. However, the solution presented should be able to

work on a geometrical level as well. We must see, then, whether geometrical objects are able to mix in the same way that physical objects are able.

Aristotle's Views on Mixing

The most explicit examination of mixing by Aristotle comes at the end of book one of De Generatione et Corruptione. There we are told

...anythng is combinable which, being readily adaptable in shape, is such as to suffer action and to act; and it is combinable with another thing similarly characterized (for the combinable is relative to the combinable); and combination is unification of the combinables, resulting from their alterati~n.~

For example, in the last chapter part of the focus was on water and wine. Both of these are adaptable in shape and can be acted upon. Thus, they can be mixed. In fact, we are told that part of the reason they are so easily combined is because they are liquids.

For instance, liquids are the most combinable of all bodies - because, of all divisible materials, the liquid is the most readily adaptable in shape, unless it be viscous. Viscous liquids, it is true, produce no effect except to increase the bulk4

The idea that viscous liquids do not combine easily should not be interpreted as saying that they do not combine at all. Aristotle gives us an example of a viscous liquid being mixed.

In the Metaphysica, Aristotle mentions that honey-water or honey-milk (pehi~parov)is a mixture (~p&ur<).~Although water and milk might be regarded as non-viscous, we would tend to think of honey as a viscous liquid. Thus though they may not combine easily, they do combine. Perhaps we might say that since viscous liquids take a greater time to lose or gain any particular shape, they are closer to being solid than the more fluid liquids. Further, we know that solids do not readily combine, because the only way that we can have a combination of solids is to juxtapose them. This means that we can put any solids we like beside one another, but we will never have a true mixture. Aristotle uses grain to illustrate this point.

When the combining constituents have been divided into parts so small, and have been juxtaposed in such a manner, that fails to discriminate them one from another, have they been combined? Or is it rather when any and every part of the constituent is juxtaposed to a part of the other? The term, no doubt, is applied in the latter sense: we speak, e.g., of wheat having been combined with barley when each grain of the one is juxtaposed to a grain of the other. But every body is divisible and therefore, since body combined with body is uniform, any and every part of each constituent ought to be juxtaposed to a part of the other.6

If this is correct, then we might say that no matter how thoroughly we might shuffle bodies 96 in a mixture, we will never come to have a truly homogeneous mixture. However, in De

Sensu, Aristotle gives us an example of mixing done by means of juxtaposition. He says there that when we have things that cannot further be divided, such as a horse or a man, we can juxtapose them in order to get a mixture consisting of both. For example, if we have a group of men and a group of horses, we cannot divide individual men so as to mix them with individual horses. However, we can put the two groups together to form a single group of men and horses.' This is further elaborated upon in the Topica. There Aristotle tells us that mixture is not always a fusion. We can mix dry things, such as grains, without saying that they have combined to form a different whole.8

Part of the problem surrounding the whole idea of mixing is a linguistic problem. In

English there is the tendency to speak in a general manner about mixing. What I mean by this is that both homogeneous and heterogeneous mixtures are called mixtures, and unless specifically called for, there is often no attempt to differentiate between the two. When working with a translation, this is especially important, as Aristotle uses two words for mixing which are often both translated in English to mean 'mixture'. One of these words is pic~s.The other is ~p&aq.Of the two, the latter is by far the stronger word. While both pi& and KP&OI.

And the being of some things will be defined by all of these qualities, because some parts of them are mixed, others are fused, others are bound together, ...g 97

This shows clearly, that although the translation may at times indicate that both pic15 and lcp&al

Of the two types of mixing, we are more concerned with KP~~Sthan with pic^^

EAristotle had accepted atomism, he could have said that when boundaries touch, they mix in exactly the way pic^^ implies - the atoms that make up the boundaries intermingle.

However, since Aristotle did not accept atomism, the boundaries of any objects are not made up of particles that can intermingle. Thus, we must have mixture that is more like the mixing of liquids than the mixing of grains. Ln other words, the problem of mixing with which we must ded involves the concept of ~pka~~.

Difficulties with Aristotle's View

1. The Problem of Coincident Bodies

It has been suggested by a number of thinkers, such as Ross, Fine, and Sharvy, that we might avoid the problem of mixing at the boundaries by taking Aristotle at his word.

When we are told that two bodies can be together (bpa) or in contact and still have their boundaries remain distinct - leaving us with two boundaries in the same place rather than one or, in the case of some continuous objects, none - we should understand this as meaning that the two objects in question simply have nothing between them. As shown in the last chapter,

I believe that this is an unacceptable answer for several reasons. Still, it does allow for an attractive solution to the problem of touching. If we say that two things are able to touch in this way, then we can have shared boundaries simply by alternating bits from each boundary.

This would be like a zipper. When the two objects are apart, they each have their own edges. 98

When they touch, the edges mix by means of an intermingling. In other words, contact implies p ict~.

The difficulty here is that we are left with the same problems we had before. We may have solved how the macroscopic objects touch, but now the bits that have intermingled are presumably touching. Are we to say that they touch each other in the same way as the larger objects, and so on down the microscopic ladder? It seems that this is, at best, a way to postpone the problem. Better, I think, to face the problem in terms of ~pka~~rather than P m.

One of the difficulties we face when we consider mixing as K~~OLSinvolves not just the idea of boundaries occupying the same place, but the seemingly impossible idea of entire bodies doing so. We know that Aristotle does not allow that two bodies can occupy the same place at the same time. He mentions this in several places, including De Generatione el

Corruptione.

If, on the other hand, [a growing thing] grows by accession of a body, there will be two bodies - that which grows and that which increases it - in the sarne place; and this too is irnpossib~e.'~

It is this idea of two bodies not being able to be in the sarne place that caused some difficulty when we examined the concept of points in whole-with-whole contact. Recall that there we said that one of the problems involved was that whole-with-whole contact between points would mean that Aristotle might have to allow for two bodies to be in the same place. If we allow for two boundaries to come into whole-with-whole contact we have the same sort of 99 problem. If, after contact, the boundaries still exist as part of the original objects, then we can say that the objects are both in different places. However, this is difficult for the concept of touching as we have seen in the last chapter. If, on the other hand, the boundaries mix so that they become one, we find a problem similar to one first introduced in our discussion of points. Specifically, if two boundaries mix, is the remaining boundary part of one object or the other, is it part of both, or is it part of neither? As before, the first option, that of the boundary belonging to one or the other, seems problematic. If we claim that the two bodies have mixed, it seems strange to then say that the resulting mixture is only part of one body.

The possibility that the boundary belongs to neither object is also problematic. Consider that this would imply that when two objects touch in such a way that they mix, they do not actually touch each other. Rather, they come into contact at a boundary made of the two objects. This idea, that by mixing objects do not touch, is not the only problem here. If we allow for it to be a possibility, we must say that the same problem exists for that which the objects do touch. Hence, when the boundaries mix, they no longer belong to either object.

Since we want to say that there is contact with the boundaries, we can look at the way in which the original objects now touch the separate boundaries. Of course, this would lead to the same sort of problem, leading to an absurd regression.

The idea that the boundary is part of both is possible in two ways, the first of which is a trivial sense. What I mean by this is that even if the boundary belongs to the form of one body, it may still act as an external boundary for the other. This is because we can say that the boundary is the form of one body and merely a bounding surface for the other. The more interesting, and certainly more problematic way to interpret this, is to say that the boundary 100

is part of each body in the sense that it belongs to the form of each. This must lead us to ask

what it might be that differentiates between a single body and two bodies with a shared

boundary. It is the solution to this that will give us a better understanding of what we mean

when we differentiate between continuity and contiguity.

2. The Problem of hstantaneous Change

If we suppose that two objects can share a boundary in this way, then we must consider the time involved. What I mean by this is that when any two things mix, there is change. Aristotle tells us that change requires time. However, this must mean that when we bring two objects into contact, they must touch fust before they can mix. If they mix at the same moment that they touch, then we have a case where change is instantaneous.

3. The Problem of Generation

A further problem arises when we consider what happens at the mixed boundary without considering the greater whole of the touching objects. When two objects of the same sort touch we might allow for the boundaries to mix since there would be nothing new created. In fact, it seems to be accepted by some, such as Fine, that contact at the boundaries is unproblematic. He writes in a discussion about mixing

But [Aristotle] cannot simply be claiming that when two mixable items are placed side by side they will mix. For interaction requires contact; and when two items are placed side by side, they will only have contact at the boundary. It therefore needs to be made clear how the parts of the bodies beyond the boundary are capable of interacting."

I do think that Fine is correct in much of what is said here. Interactions such as mixing do 101 require contact, and when we look at how it is that bodies mix we need to understand how the entire system reacts, not merely how the boundaries react. Also, Fine seems to be correct when he says that merely being placed side by side is not sufficient for mixing. As we have seen earlier Aristotle explicitly tells us that succession, which fulfils the concept of being side by side, is not sufficient for contact. In order to be contiguous we need both succession and contact. What Fine neglects is that contact at the boundaries is as problematic as mixing beyond the boundaries.

Aristotle tells us that

Things are said to be together in place when they are in one primary place and to be apart when they are in different places. Things are said to be in contact when their extremities are together.''

Thus, contact requires the boundaries of two objects to be together. Further, Aristotle tells us that extremities can be together without being one." The difficulty this presents is one that we have already seen: if the extremities are understood as being in different places, then they cannot be together. We might solve this by saying that when we are told that two boundaries in contact are not one, what we should understand is that these boundaries are separable into the original boundaries we had prior to contact. This is unlike two objects which become continuous, as in this case we would say that the boundary, as it is understood to exist in a continuous object, is not immediately separable into two boundaries.

Part of the problem here deals with what Fine calls 'tracking'. If we have two bodies that mix and we wish to separate them again, how are we to find the parts which belonged 102 to each of the original bodies? If two bodies are to be in contact but not continuous, there must be a way to separate them. The difficulty here is that when we have mixing at the boundaries ,we have something between our two touching bodies which is made of neither type of matter. It is neither copper nor tin. It is bronze. If this theory of mixing at the boundaries is to work, we must have some method to separate bodies once their boundaries have mixed.

Further, assume that the boundaries of two different objects are allowed to mix, and that we are left with a situation whereby we have two different objects joined by a third type of stuff. For example, if a brick of copper touches a brick of tin and their boundaries mix, then we have each of these two bricks touching bronze. Although this might seem strange it is, in fact, related to the problem of coincident boundaries. Recall that there we said that the boundaries must belong to some object, probably both, since to say that the objects in contact touch the boundaries which in turn touch the other object was odd. But we have that very situation here. A copper brick which touches a tin one in this way seems to be actually touching a boundary of bronze. It is the bronze boundary that touches the tin. But this suggests that contact between dissimilar objects is not possible, since there will always be a third type of matter between them.

4. The Problem of Geometrical Mixing

Up until this point, I have confined the discussion to the problem of physical touching. As discussed earlier, however, we know that mixing as it applies to the geometrical problem of touching also needs to be examined. The reason that I have avoided 103 it until now is that Aristotle does not seem to discuss mixing with regard to geometry. While not altogether strange, given that mixing really does seem to apply more readily to the physical world, it is important that we not overlook this aspect of the theory. If the solutions presented are to work, they must apply for both the physical and the geometrical. Because of this, I intend to End a solution to this difficulty before proceeding to the others. By doing so, we will be able to apply to the geometrical the solutions which we find for the physical.

Intelli~ibleMatter - Solving the Problem of Geometrical Mixing

Before proceeding, it is again important to digress. When we consider the problem of mixing we are dealing with Aristotle's theory of matter. However, since we are not only concerned with physical mixing, but also with geometrical mixing, we need to understand something about the matter of geometry. Aristotle was aware that there are differences between the physical and mathematical worlds. When considering how each is studied, he says in the early passages of the Physica

The next point to consider is how the mathematician differs from the student of nature; for natural bodies contain surfaces and volumes, lines and points, and these are the subject-matter of mathematics. l4

Shortly after, he defines the difference between physics and mathematics. l5 He suggests that mathematics studies things such as lines, planes, and points, while physics studies bodies which incidentally have in them these objects. Likewise, in De Caeld6, we are told that a physicist studies objects constituted in addition, while the mathematician looks at objects that have been abstracted. This seems to imply that the objects of mathematics exist in the

objects of the physical world. However, Aristotle makes it very clear in Metaphysics M2

that mathematical objects cannot exist in sensible things1' and that they also cannot exist

separately from sensible things.18 Moreover, when Aristotle discusses the possibility in book

I' of something intermediate between Forms and perceptible things. he writes

Further, it follows from this theory that there are two solids in the same place, and that intermediates are not immovable, since they are in the moving perceptible things. And in general to what purpose would one suppose them to exist, but to exist in perceptible things? For the same paradoxical results will follow which we have already mentioned; there will be a heaven besides the heaven, only it will be not apart but in the same place; which is still more impossible. l9

If this is unresolved, then in order to rid ourselves of trying to find how geometrical objects exist we may be forced to say that the objects of mathematics do not exist at all. This poses a difficulty, as we want the objects of mathematics to exist in some way, since we do not want to be forced into the position of claiming that mathematics is about nothing. However, if they can neither exist apart from physical objects nor exist in these objects, then it seems that we are Ieft with objects which have no place to exist. A solution to this is suggested by

Feyerabend. He notes that mathematics deals with things that are not separate, but are dealt with as things that can be separated. This leads him to resolve the contradiction by pointing out that Aristotle often tells us "that things are said to be in many different ways." In other 105 words, mathematical objects have separate existence in some senses but not in others."

Thus, when we consider the objects of mathematics as ideal geometrical objects, they are separate from the physical. However, we are still abie to find instances of these objects in the physical world, as when we find a triangle in a bronze triangle. This seems a reasonable, if somewhat simple approach, and requires a more detailed examination.

We might try to settte the difficulty by simply taking Aristotle's assertion at face . He writes

Tb 6L Bv hdyeta~pEv rrohla~Q5,hhlb xpb~&v ~ai piav nvh @u'o~v~ai oljx 6pwv6poq...

There are many senses in which a thing may be said to 'be', but they are related to one central point, one definite kind of thing, and are not hornonyrno~s.~'

Aristotle is here expounding his idea of "focal meaning." He illustrates this by telling us that all things healthy are related to health. For example, great endurance in a person and the medicine prescribed by a doctor are both healthy, the former because it is a preservation of health, the latter because it produces health. In this respect, we might say that when we are dealing with the objects of geometry, we are dealing with objects under a description. Just as great endurance might be said to be both healthy and athletic, we might say, for example, that a ball is both a perceptibIe object and a geometric sphere. In this way, we can have the object of geometry existing as part of the ball without having two objects exist in the same place. This still leaves us with the difficulty of understanding how it is that mathematical objects cannot exist separately from the perceptible, and yet also cannot exist within them.

Mueller also recognizes that there is a problem understanding how mathematical objects exist. He claims that while the objects of geometry cannot redly exist for Aristotle separately from physical bodies, it is clear that they are considered separately. As well, there is a significant difference between separating mathematical objects from physical bodies and treating physical bodies as mathematical objects." What primarily differentiates the objects of physics from those of mathematics is the fact that the mathematician studies physical objects in the abstract. That is to say that the mathematician does not examine these objects as inseparable parts of the physical world, but instead looks at them as ideal objects which are separate from the matter of the physical world. This way they do not exist separately from sensible things, but at the same time do not constitute a second substance within a sensible object. This does not mean that the objects of geometry are without matter.

Aristotle is clear that mathematical objects have matter, but of a different sort. In the

Metaphysica Aristotle remarks that

... some matter is sensible and some intelligible, sensible matter being for instance bronze and wood and all matter that is changeable, and intelligible matter being that which is present in sensible things not qua sensible, i.e. in the objects of mathernati~s.~~

From this passage, along with others from the ~eta~hysica'~and De ~nima", Ross concludes that the matter of geometrical objects is pure extension. It is intelligible, but is neither sensible nor physical.26 This intelligible matter is discovered by a process of abstraction (dr@ctipeoy).Ross writes

From any sensible thing you may think away the whole sensible matter. In the case of terrestrial things you can abstract from their possession of the fundamental qualities - heat or cold, dryness or fluidity - and of all the consequential qualities; in the case of celestial things you may abstract from their capacity for rotation; both alike will still have shape and size. You will have passed by abstraction from actual bodies to the objects of mathematics."

This would seem to indicate that intelligible matter (CAT voqrfl) is what is left of sensible matter once all sensible qualities are removed. It is matter in its most abstract form. This is a difficult concept to deal with, as we would think that if we removed all but extension from matter then we would be left only with space or place.

Annas raises a further problem with the notion of intelligible matter merely being abstract matterF8 She points out that we cannot simply be abstracting entirely from matter, as this is too close to the Platonic notion of Ideas which Aristotle rejected. However, the idea that when we are dealing with the objects of mathematics and geometry we are also dealing with a different kind of matter is also troublesome. This is mainly because Aristotle never clarifies exactly what this matter is in the same way that he does for 'standard' matter.

Additionally, during one of his discussions about matter Aristotle writes,

For if this is not substance, it is beyond us to say what else is. When all else is taken away evidently nothing but matter remains. For of the other elements some are affections, products, and capacities of bodies, while length, breadth, and depth are quantities and not substances. For a quantity is not a substance; but the substance is rather that to which these belong primarily. But when length and breadth and depth are taken away we see nothing left except that which is bounded by these, whatever it be; so that to those who consider the 108

question thus matter alone must seem to be substance.2g

This suggests that after abstraction we must be left with matter of some sort in order to have geometrical objects to consider. Gaukroger tells us that we need this matter because the shapes that we are left with after abstraction must be the shapes of something else.30

Abstraction in a certain way gives us length in all three dimensions. Thus we can learn about the line, the plane, and the solid." However, it does not tell us what sort of matter is left after abstraction, save that it must be intelligible in some way.

The difficulty with this idea of being left, after abstraction, with intelligible matter having a certain form, is that it becomes difficuIt to determine what differentiates the intelligible from the sensible. Gaukroger writes

Because we abstract these forms, the form of what we abstract from and the form that we abstract are the same: the sensible circle and the noetic circle have the same form. This leads us to ask what it is that distinguishes the noetic circle from the sensible circle.32

Gaukroger believes that when we abstract we are left with a substratum. This, as we have already seen, is spatial extension. Since spatial extension is an abstraction, it does not exist independently. As well, it is not sensible since all sensible qualities have been removed from it. Thus, the intelligible object differs from the sensible one in virtue of the abstraction used to find it.

If we hold with the interpretation of intelligible matter that has been put forth, we must still resolve exactly what abstraction is and how it is to take place. Owens tells us that abstraction is merely a process of subtraction for Aristotle. We remove those qualities which we do not want to consider, and retain what has been separated from these qualities." We find Aristotle giving this definition when he explains the process of abstraction in the

Now the mathematician, though he too treats of these things, nevertheless does not treat of them as the limits of a natural body; nor does he consider the attributes indicated as the attributes of such bodies. That is why he separates them; for in thought they are separable from motion, and it makes no difference, nor does any falsity result, if they are separated. The holders of the theory of Forms do the same, though they are not aware of it; for they separate the objects of natural science, which are less separable than those of mathematics. This becomes plain if one tries to state in each of the two cases the definitions of the things and their attributes. Odd and even, straight and curved, and likewise number, line, and figure, do not involve motion; not so flesh and bone and man - these are defined like snub nose, not Iike curved. Similar evidence is supplied by the more natural of the branches of mathematics, such as optics, harmonics, and astronomy. These are in a way the converse of geometry. While geometry investigates natural lines but not qw natural, optics investigates mathematical lines, but qua natural, not qua mathematical?

Lear notes several features of Aristotle's phiiosophy of mathematics in this passage. Among these is the notion that the mathematician is able to separate in thought surfaces, volumes, lengths, and points from the physical bodies that contain therna3' This idea of separation in thought is crucial if we are to retain the notion that mathematical objects cannot exist on their own. Essentially, the idea of separation amounts to the following.

... mathematical properties are truly instantiated in physical objects and, by applying a predicate filter [a means Lear uses to show how certain properties can be separated from a whole], we can consider these objects as solely instantiating the appropriate proper tie^.^^

The 'predicate filter' to which Lear refers is merely a precise way of omitting certain qualities while retaining others. At its core, this manner of filtering mathematical properties can be expressed, as it is expressed by Aristotle, as a removal of all sensible qualities from a physical object.

...and since, as the mathematician investigates abstractions (for in his investigation he eliminates ail the sensible qualities, e.g. weight and lightness, hardness and its contrary, and also heat and cold and the other sensible contrarieties, and leaves only the quantitative and continuous, and does not consider them in any other respect, and examines the relative positions of some and the consequences of these, and the commensurability and incommensurability of others, and the ratios of others; but yet we say there is one and the same science of all these things - geometry)... "

This is further expressed later in the Metaphysics Just as the part of mathematics deals not with objects which exist separately, apart from magnitudes and from numbers, but with magnitudes and numbers, not however qua such as to have magnitude or to be divisible, clearly it is possible that there should also be both formulae and demonstrations about sensible magnitudes, not however qua sensible but qua possessed of certain definite qualities. For as there are many formulae about things merely considered as in motion, apart from the essence of each such thing from their accidents, and as it is not therefore necessary that there should be either something in motion separate from sensibles, or a separate substance in the sensibles, so too in the case of moving things there will be formulae and sciences which treat them not qua moving but only qua bodies, or again only qua lines, or qua divisibles, or qua indivisibles having position, or only qua indivisibles. Thus since it is true to say without qualification that not only things which are separable but also things which are inseparable exist - for instance, that moving things exist, - it is true also to say, without qualification, that the objects of mathematics exist, and with the character ascribed to them by mathematicians. And it is true to say of the other sciences too, without qualification, that they deal with such and such a subject - not with what is accidental to it (e-g. not with the white, if the white thing is healthy, and the science has the healthy as its subject), but with that which is the subject of each science - with the healthy if it treats things qua healthy, with man if qua man. So too is it with geometry; if its subjects happen to be sensible, though it does not treat them qua sensible, the mathematical sciences will not for that reason be sciences of sensibles - nor, on the other hand, of other things separate from sensibles .38

Barnes notes that this amounts to an analogy with the non-mathematical sciences. Just as medicine deals with the healthy as its subject, so too does mathematics deal with, for example, the countable. This does not imply that the countable exists apart from a body any more than the healthy does. It is simply a way of looking at a body. It is studied qua healthy or qua countable. In this way, we no longer have to concern ourselves with the objects of geometry being separate abstract objects. While they are abstract in the sense that they are only "spoken of in virtue of abstraction", the kind of abstraction used is what Barnes refers to as "the abstraction introduced by the qua locution." This, he claims, is not what we would normally regard as abstract, and is therefore not problernati~.'~

Mueller has a different interpretation. Taking note of the passages from the

Metaphysica, he believes that the previous passage can be seen as saying that when the mathematician studies an object, a man for example, he does not study it as a man. Instead. he ignores all the sensible properties of the man. Just as Owens claimed earlier, this ignoring of properties is what Mueller considers abstraction. All that remains once this process is complete is an object which is both quantitative and continuous in three dimensions.

Mueller comes to the concIusion that

There is, then, at least an initial plausibility in supposi~gAristotle to have entertained a conception of mathematical objects, not as matterless properties, but as substance-like individuals with a special matter - intelligible matter?

There is not much difference between this interpretation and the one presented by Barnes. 113

The main difference, it seems, is in the description. In practice both interpretations involve abstraction as a means of ignoring certain properties in a body while examining others. As well, they are both in agreement that the geometrical objects do not exist without the sensible. Rather, they may be contemplated as objects of imagination, but have matter only when part of a body. Ross worries about this differentiation between physical and mathematid matter. He claims that if we accept this distinction between the types of matter involved with mathematics and physics, then we find a difficulty when we consider the 'more physical parts of mathematics' such as astronomy or ." This does not seem to be a great problem according to the interpretationsjust presented. Once abstracted from a body, the matter possessed by geometrical objects is intelligible, as one cannot entirely remove matter from a substance. However, it seems to exist for the sole purpose of allowing abstract contemplation.

Annas claims that Aristotle thinks of the objects of mathematics in the same way that he thinks of time and place. In particular, they do not exist as independent things. Rather, they depend upon the existence of other things in order to exist." Aristotle is clear, though, that mathematical objects, whatever else they may be, are abstractions from the physical.

However, these abstractions are not intended to commit Aristotle to the existence of abstract objectd3 The process of abstraction is one of method and not of matter. Annas also points out that Aristotle's treatment of mathematics in M3 is sketchy and at times incoherent and inconsistent. Given our knowledge of the fact that the problems of mathematics with which

Aristotle was coping were only in their early stages, I think that we could expect little else.

Thus, while we may not like the fact that the treatment of the objects and methods of mathematics is not as complete as the treatments of space and time, we cannot reasonably hold that every element of the theory should have been clearly defined.

Annas gives still further support to the idea that abstraction is nothing but a method of controlled ignoring. She says, after Sextus Empiricus, that

Abstraction ...thus comes down to (deliberate) lack of attention?

She further supports this with a passage from De Memuria.

Without an image thinking is impossible. For there is in such an activity an affection identical with one in geometrical demonstrations. For there is in the latter case, though we do not make any use of the fact that the quantity in the triangle is determinate, we nevertheless draw it determinate in quantity. So likewise when one thinks, although the object may not be quantitative, one envisions it as quantitative, though he thinks of it in abstraction from quantity; while on the other hand, if it is something by nature quantitative but indeterminate, one envisages it as if it had determinate quantity, though one thinks of it only as a quantity."

This seems to indicate that Aristotle was well aware of the fact that he was dealing with two different types of objects; those with concrete, determinate qualities, and those with abstract qualities. Lear says something similar. He believes that the fiction of calling a line a foot long when it is not works because geometers are concerned with the geometrical properties which follow from a given object. To think of a geometrical object as having properties such as definite length, colour, or shape is, by Aristotle's thinking, a confusion, as by doing this one is not considering a as separated from all irrelevant properties.46

It is important to remember that mathematical objects are abstractions from the physical, which means that the study of the objects of geometry is really the study of a specific aspect of the physical world. Thus, unlike Plato's later theories, where mathematics held a more important role than physics. Aristotle holds that mathematics studies the objects of physics using the tool of abstraction.

Each question will be best investigated in this way - by supposing separate what is not separate, as the arithmetician and the geometer do. For a man qua man is one indivisible thing; and the arithmetician supposes one indivisible thing, and then considers whether any attribute belongs to a man qua indivisible. But the geometer treats him neither qua man nor qua indivisible, but as a solid. For evidently the attributes which would have belonged to him even if he had not been indivisible, can belong to him apart from these attributes. Thus, then, geometers speak correctly - they talk about existing things, and their subjects do exist; for being has two forms - it exists not only in fulfilment but also as matter."

In other words, geometry deals only with a specific part of physics, ignoring all those parts that do not involve geometry. Note that this does not preclude treating the objects of geometry as physical objects. In effect. it allows us to say that when we deal with geometry we are simply dealing with a subset of the physical.

The DiE~cultvof Making the Transition from Physical to Geometrical Obiects

Despite the assertion that the objects of mathematics are merely derived in some sense from the physical, we must still deal with some difficulties. Aristotle tells us that there are some important differences between the two realms.

And astronomy also cannot be dealing with perceptible magnitudes nor with this heaven above us. For neither are perceptible lines such lines as the geometer speaks of (for no perceptible thing is straight or curved in this way; for a hoop touches a straight edge not at a point, but as Protagoras said it did, in his refutation of the geometers), nor are the movements and complex in the heavens like those of which astronomy treats, nor have geometrical points the same nature as actual stars.48

Aristotle seems to be telling us here that physical, sensible objects do not act in the same manner as we would have the ideal objects of mathematics act. 'This Aristotie observes with the question, "With what sort of things must the mathematician be supposed to deal?

Certainly not with the things around us. For none of these is like what the mathematical sciences investigate." A bronze sphere (even as solid) does not touch an iron bar (even as plane or line) in a pointmdgAristotle is not alone among the ancient Greeks in thinking this.

It seems to be the case that the Pythagoreans believed this of physical tangency as well.''

However, geometry tells us that a straight edge does touch a circle at a point. If the objects of mathematics are abstractions, it would seem that the physical objects should behave at least in some respects as the mathematical ones do. If Mueller is correct, then we have a Leardoes not agree with Mueller's interpretation. He tells us that Aristotle would say that Protagoras is correct only in that the hoops found in nature are not perfectly round, nor are physical lines perfectly straight. Thus, there is no difficulty in saying that they do not touch at a point. However, Lear also notes that

...Aristotle is not committed to saying that there are any perfectly circular hoops existing in the world. All Aristotle must say is : i) insofar as a hoop is a circle it wilI touch a straight edge at a point; ii) there are some physical substances that are circular. (Such circular substances need not be hoops.) Claim (i) is true: inasmuch as a hoop fails to touch a straight edge at a point, thus much does it fail to be a circle. And there is certainly evidence that Aristotle believed claim (ii). Of course, Aristotle thought that the stars were spheres and that they moved in circular orbits. But there is dso evidence that he thought that even in the sublunary world physical objects could perfectly instantiate geometrical properties?

Also, Lear points to a passage in De ~nirna~'that shows both that Aristotle believed that there were physical objects with perfectly straight edges and that when dealing with such perfect objects we can have a situation where an iron bar can touch a bronze sphere at a point.53 If this is correct, then we can make the claim that when a geometer looks at a straight edge touching a hoop, he can make the claim that they touch at a point in exactly the same way that he makes the claim that a line drawn in the sand is a given length even though neither is the case. The geometer is concerned not with the particular size or shape to be found in the object or drawing before him. Along with abstracting colour, size, and other irrelevant qualities, he can ignore the imperfections in the actual model so as to consider the desired ideal. Put another way, the objects of geometry can be thought of as sensible, but this is only an . The geometer does not treat them as sensible objects."

One problem with this treatment of the objects of geometry is that Aristotle seems 118 to indicate in Metaphysica 2, that geometrical objects are not substances. He mentions those who believe that they are substances (a probable reference to the Pythagoreans) among those who he believes are wrong about the nature of substan~e.~~Also, in Metaphysica H, he writes

Some things are or are not without coming to be or ceasing to be, for instance points (if they exist) and forms generall~.'~

Following this he says that such things as do not come to be or cease to be do not have matter. This would imply that points do not have matter. Perhaps this can be taken as a reference only to 'normal' matter and not to the aforementioned abstract matter.

This having been said, how does the problem of touching move from the physical to the geometrical? We have already seen that Aristotle claims that a physical straight edge may not touch a hoop at a single point. We can easily concede this if we follow Leafs interpretation. The difficulty comes when we try to assess the problem with objects that fulfil the conditions which Lear sets out: if we do have a straight edge touching a perfectly circular object, then do they - in fact can they - touch at a single point? Recall that Lear claims that given the proper shapes - a solid that is actually circular and an edge that is actually straight - we will have the two objects touching at a point. But this brings us back to the initial problem. Since these points are separated from all irrelevant qualities when we consider them as mere geometrical entities we need worry about nothing but whether the point is a single one shared between the two objects or whether there is more than one point involved. Nonetheless, since all geometrical objects are composed of intelligible matter at 119 most, it seems to be only an intellectual problem. However, when we look at the problem without abstraction we are faced with serious difficulties. If the two objects touch at a single point, is that point part of the hoop or the edge? If both objects share the point, is it made of the material of one or the other? Essentially, all the problems of touching once again come back to haunt us.

hasraises a further problem with the notion of abstract matter. As I have already noted, she points out that we cannot simply be abstracting entirely from matter, as this is too close to the Platonic notion of Ideas that Aristotle rejected." However, the idea that when we are dealing with the objects of mathematics and geometry we are also dealing with a different kind of matter is also troublesome. This is mainly because Aristotle never clarifies exactly what this matter is in the same way that he does for 'standard' matter? Even with these difficulties, Evans believes that we have enough information to tell us what this second kind of matter is. He says that the mathematician is concerned with geometrical shapes which are found in material substances. Further, there is not a concern with concepts such as 'straightness' or 'circularity'. The geometer deals with the more concrete notions of 'what is straight' or 'what is circular' - notions which are particular and must exist in a material element. The element that these notions exist in is intelligible matter."

Returning to the poict of this section, we can now sum up the difficulties involved in making the move from the physical to the geometrical. We must realize that the objects of mathematics are abstractions of a sort. Specifically, though there may not be perfect examples of any given geometrical object before us, we can understand that such objects exist. We must take care, though, to avoid falling into the trap of saying that these objects 120 are the standards by which all others of the same type are measured. To do so would be akin to asserting something like Plato's theory of Forms, and would be incorrect in an Aristotelian context. What must be realized is that abstraction is a method used to find geometrical forms which exist in sensible matter as opposed to noetic matter. The geometrical objects are such that when we look for them by means of abstraction, we are not really finding something different from the object from which we abstract. We are simply examining one aspect of the object by itself. Further, when we consider a geometrical object in the absence of a physical object, just as when we consider an object which exists only in thought, we are considering a particular image. In our thought it is a definite quantity or particular image.

This means that it can be considered to behave in the same way as an abstraction from a physical object, which also exists in thought in this way." Thus, we can now understand how it is that the geometrical problem of touching applies to the physical problem.

Solving;- the Other Difficulties

Of the four problems presented at the outset of this chapter, only one has been dealt with. While it may be the case that notions of abstraction and intelligible matter will be useful in solving the difficulties associated with physical mixing, on their own they do not do so. Thus, we have only shown that the geometrical problem can be solved for Aristotle provided that the physical problem can be solved. And in order to solve the physical problem there is one further topic which needs to be examined: we need to look at the distinction between potential and actual. 1. Physica 227a2 1-23.

2. Physica 226b27-3 1.

3. De Generatione et Corruptione 328b20-24.

4. De Generatione et Corruptione 328b3-5.

5. Metaphysica 1042b16-17.

6. De Generatione et Corruptione 327b34-328a5.

7. De Sensu 440b1-13.

9. Metaphysica 1042b28-30.

10. De Generatione et Corruptione 32 la8- 10.

11. Kit Fine, "The Problem of Mixture," Pacific Philosophical Quarterly 76 ( 1995): 355.

12. Physica 226b2 1-23.

13. Physica 227a23.

14. Physica 193b23-25.

15. Physica 193b30.

16. De Caelo 299a16-17.

17. Metaphysica 1076a38-b 12.

18. Metaphysica lO76b 12-38.

19. Metaphysica 998a13-19.

20. Paul Feyerabend, "Some Observations on Aristotle's Theory of Mathematics and of the Continuum," Midwest Studies in Philosophy VIII (1983): 67-68.

2 1. Metaphysica 1003a32-33.

22. Ian Mueller, "Aristotle on Geometrical Objects," Archiv fir der Geschichfe der Phiiosophie 52 (1970): 159.

23. Metaphysica 1036a9-12. 24. Metaphysica 1036b32ff

25. De Animn 403b17.

26. Sir David Ross, Aristofle (New York, Routledge, 1995) 72.

27. Ross 174.

28. Julia hnas, Metaphysics M arid N (Oxford, Clarendon Press, 1976) 33.

29. Metaphysica 1029a10- 19.

30. Stephen Gaukroger, "Aristotle on Intelligible Matter," 25 (1980): 188.

31. Mueiler 165-166.

32. Gaukroger 188.

33. Joseph Owens, The Doctrine of Being in the Aristotelian Metaphysics (Toronto, Pontifical Institute of Mediaeval Studies, 1978) 382.

35. Jonathan Lear, "Aristotle's Philosophy of Mathematics," Philosophical Review XCI (1982): 163.

36. Lear 170.

37. Metaphysica 1061 a28-b3.

38. Metaphysica lOVb 17- lO78aS.

39. Jonathan Barnes, "Metaphysics", The Cambridge Companion to Aristotle, ed. Jonathan Barnes (Cambridge University Press, 1996) 85-87.

40. Mueller 164.

41. Ross 72.

42. Annas 28.

43. Annas 30.

44. Annas 32. 46. Lear 173ff.

47. Metaphysica 1078a2 1-3 1.

48. Metaphysica 997b34-998a6.

49. Mueller 158.

50. Paul Tannery, La Gkomdtrie Grecque (New York, Arno Press, 1976) 123.

5 1. Lear 176- 177.

52. De Aninta 403al0-16.

54. Feyerabend 68.

55. David Bostok, Aristotle: Metaphysics, Books Zand H (Oxford, Clarendon Press, L994) 70.

56. Metaphysica 1044b21-22.

59. Melbourne G. Evans, The Physical Philosophy of Arisrotle (Albuquerque, The University of New Mexico Press, 1964) 42.

60. Ross 152. Chapter 5 - Actuality and Potentialitv

Connecting Intelligible Matter to the ActualPotential Distinction The last chapter dealt with the concept of intelligible matter as formal cause. Nothing, though, has been said about the formal cause of an object since the discussion of boundaries. This section reintroduces the concept of form to the discussion and relates it to the concepts of actuality and potentiality.

Explaining Actualitv and Potentialitv Some of the solutions presented up to this point deal in some way with the concepts of actuality and potentiality. This section explains these concepts and how they are related.

Actualization and Geometry The concept of actuality can now be adduced to help solve the problem of touching. This is done first by returning to the geometrical problem. This involves seeing how points and geometrical objects can be related to intelligible matter by means of actualizing potentials.

Actualizing Potentials The method of actualizing potentids is explored.

Solving- the Problems With Boundaries The problems encountered in the chapter on boundaries are reexamined. These are then solved by using the actual/potentiaI distinction.

Solving- the Problems of Coincident Boundaries, Instantaneous Change. and Generation In the last chapter there were four problems introduced. However, only one was solved using the concept of intelligible matter alone. Now, with the added concept of actualization, we can return to these problems and offer solutions. Connecting IntelIi~ibleMatter to the Actual/Potential Distinction

A significant part of the last chapter dealt with the concept of intelligible matter, and it did so with a view to solving the problems involved with geometrical mixing. However, all that this was able to do for us at that point was to make the connection between the physical and the geometrical. It allowed us to show that since there is an explicit connection between geometrical objects and physical objects - that connection being intelligible matter - the same solutions which apply to the physical should be able to be applied to the geometrical. Though this is not itself the solution to the problem of contact, it is an important step to be made in that direction. Recall that the first families of solutions that were examined, all of which were problematic in one way or another, were specifically geometrical in nature. By making an explicit connection between the physics and geometry we should be able to use physical results to help find solutions to the geomevicai problems.

Despite the fact that the concept of intelligible matter will not solve all our problems by itself, it is of greater importance than suggested in the last chapter. One of the things that was mentioned was that geometrical objects are not composed of any matter other than intelligible matter. However, it is clear that it is not simply matter which makes a thing what it is. Bricks may make a house, but a pile of bricks is not a house. Thus, in Metaphysica 2,

Aristotle writes The object of inquiry is most overlooked where one term is not expressly predicated of another (e.g when we enquire why man is), because we do not distinguish the elements before we begin to inquire; if not, it is not clear whether the inquiry is significant or unmeaning. Since we must know the existence of the thing and it must be given, clearly the question is why the matter is some individual thing, e.g. why are these materials a house? Because that which was the essence of 2 house is present. And why is this individual thing, or this body in this state, a man? Therefore, what we seek is the cause, i.e. the form, by reason of which the matter is some definite thing; and this is the substance of the thing.'

This suggests that when we consider any object, we must look beyond the matter which makes up the object and examine the form as well. Thus, while geometrical objects may have intelligible matter as a material cause, we must look beyond this to find their formal cause. This is not an unanticipated move. In Chapter 3 on boundaries, I was concerned with definitions that allowed us to deal with boundaries as both the last point of a thing as and the form of a thing. Now, it seems that we have reason to make a connection between form and a specific type of matter.

'Form1and 'matter' are, for Aristotle, often interchangeable with the terms 'shape' and

'stuff. So we understand 'dbo~'as meaning both 'form' and 'shape', and 'Chq' as meaning both 'matter' and 'stuff. Put another way, the matter of a thing is the stuff it is made of, while the form is the shape the stuff takes. We are essentially told that substances are composite entities, being made up of both form and matterm2Matter has the potentiality to be acted upon and to receive certain shapes. The form is that shape which exists in the matter at the end of production. Thus, after the production of an object we have actually what before we had only p~tentially.~The example we see AristotIe using to explain actuality can serve to make this clearer.

Actuality means the existence of the thing, not in the way we express by 'potentially'; we say that potentially, for instance, a statue of Hermes in the block of wood and the half-line is in the whole, because it might be separated out, and even the man who is not studying we call a man of science, if he is capable of studying. Otherwise, act~ally.~

Thus, the image of Hermes exists potentially in the block of wood. Likewise, so does the image of Zeus. Once the wood is carved into the image of Hermes, this image actually exists. Likewise, prior to drawing a line, the line exists potentially. Once drawn, it exists actually. The matter in which it exists, however, is that matter which is left after abstraction

- intelligible matter.

Ex~lainingActuditv and Potentialitv

Many of the possible resolutions to the initial problem suggested up until now have dealt in some manner with the notions of actuality and potentiality. If the theory I have presented is correct, then we know that points exist in some manner, but exactly how they exist is unclear. Whether their existence is merely a matter of geometry or whether there is a more 'substantial' existence involved, all we can say, besides that there is probably an aspect of intelligible matter related to their existence, is that there seems to be some distinction which Aristotle can make between the real and ideal worlds. Translated into terms which would be more familiar to an Aristotelian, this involves saying that resolution 128 to the problems we have encountered with points rests in the distinction between the actual and the potential. Similarly, the answer to why continuity and contiguity are not the same thing may rest in this distinction as well. To elaborate, we can say that boundaries exist, but whether they are between two contiguous objects or part of a continuum depends upon whether they exist actually or potentially.

This dependence on the actual/potential distinction is not without difficulties.

Essentially we are saying that when we consider geometrical objects, in particular when we consider points and boundaries, we actualize potentials. Recall, for example that in the earlier discussion of points the idea was put forth that points exist everywhere in an object, but only potentially. When we make a selection or a division we actudize a point, line, etc.

This fixes certain states of affairs, so that any later decisions are bound by the results of the earlier . This seems to give space a strange, almost dynamic, appearance. We might go so far as to claim that if this interpretation is correct, then Aristotle gives space a quality which we might call "Heisenberg-ian", insofar as it can be changed by observation. Of course, this language would be carrying things too far. Through all of this we must bear in mind that whatever other ideas AristotIe might or might not have held, he was adamant that there was no such thing as void or empty space. At most, we might be able to say that we are actualizing potentials in intelligible matter. Thus, a given space might appear empty to our eyes, but there is really something there. This means that we could not call space dynamic. However, it might imply that matter has a dynamic element which has been overlooked. Further, this notion that intelligible matter is dynamic must also be considered in light of the theory that it is at the heart of abstraction to geometrical objects. In De Generatione Animalium Aristotle tells us about how things are created.

But how is each part formed? We must answer this by starting in the first instance from the principle that, in all products of nature or art, a thing is made by something actually existing out of that which is potentially such as the finished product.'

If we are to take this passage at face value, then it seems to tell us that when anything is created, it comes from a state in which it existed only potentially. As well, in the

Metaphysics we are told

And the definition of that which as a result of thought comes to be in fulfilment from having been potentially is that when it has been wished it comes to pass if nothing external hinders it, while the condition on the other side - viz. in that which is healed - is that nothing hinders the result. Similarly, there is potentially a house, if nothing in the thing acted on - i.e. in the matter - prevents it from becoming a house, and if there is nothing which must be added or taken away or changed; this is potentially a house, and the same is true of all other things for which the source of their becoming is e~ternal.~

This passage is important as, according to ROSS', it is here that Aristotle is trying to determine the conditions under which any object may be said to be potentially a second object. This passage, however, deals with artistic production, so the objects in question must be acted upon by some outside force. It seems to me that this should also apply to the geometrical problem at hand, especially given Aristotle's remarks in the Metaphysica concerning substance as it applies to a sculpted figure.

Besides this, no sort of shape is present in the solid more than any other; so that if Hennes is not in the stone, neither is half of the cube in the cube as something determinate; therefore the surface is in not in it either; for if any sort of surface were in it, the surface which marks off half the of the cube would be in it too. And the same account applies to the line and the point and the unit. Therefore, if on the one hand body is in the highest degree substance, and on the other hand these things are so more than body, but these are not even instances of substance, it baffles us to say what being is and what the substance of things ism8

This passage expIicit1y tells us that the same reasoning that is used with the physical can be used with the geometrical. Aristotle first makes a claim about a figure in a piece of stone.

The theory is not that the figure, as some sculptors might say, is in the stone and one merely has to remove everything that does not belong. Rather, the figure does not exist until it is actually carved. This is related to the claim that when we examine a cube, we cannot find half of the cube in it until we make a division. If the figure of the half-cube were to exist prior to this, then so would the limit of half of the cube, and so would the limit of the half of the half, and so on to infinity. This must mean that the divisions exist only potentially, not actually. Further, there is a connection made here between the mathematical and the physical which is even more explicit. Since Aristotle is discussing solid figures here, we might be inclined to think that he is discussing only those figures that can be represented physically. However, just as this argument can be made for the limits of solids, which can be construed as surfaces, Aristotle claims that we can make similar arguments for geometrical objects with fewer dimensions, namely Lines, points, and even units. Therefore, since Aristotle makes such an explicit link between the act of sculpting a physical object and the act of dividing a geometrical one, at least as the concepts of actuality and potentiality are concerned, we are probably safe in applying the same methods used in artistic production to those used in geometrical production.

Returning to the passage at hand, the assumption made by Ross concerning conditions under which some object may be potentially another is confirmed by Aquinas.

And the same holds true in the case of other things whether their principle of perfection is outside of them, as is the case of artificial things, or within them, as is the case of natural things. And they are always in potency to actuality when they can be brought to actuality without any external thing hindering them. However, seed is not such, for an animal must be produced from it through many changes; but when by its proper active principle, i.e., something in a state of actuality, it can already become such, it is then already in potency?

Though the passage deals with artistic production, we have just seen that it can probably be applied to geometrical production as well. As I have argued it, the point exists potentially in a place, but is only actualized once a decision has been made to make a division or choose a point. This would make intelligible matter the matter which has the potential, the geometer the artist, and the point the product. Note that this fits with the theory that space, construed as intelligible matter, is .mutable. Essentially the argument can be made that the basic elements of geometry are potentially in space. Since they are basic, they do not require any intermediate stage in order to be produced actually. This is especially true for points. It may 132 not always hold for figures with any sort of dimension, such as lines or planes, as they can be produced through the movement of points. However, if we consider a line as the division of a plane and not as the movement of a point through space, then we might say that lines are potentially in space just as points are. Thus, if we can apply this passage to geometry, then what has been suggested earlier, namely that the resolution to the problem of geometrical objects touching one another is rooted in actuality and potentiality, seems to be aU the more plausible. However, in order for us to be sure that this can work we must further examine the concept of actualization with respect to geometrical objects.

Actualization and Geometw

In the Physica we are told

...iso0 ~UV@JE~ dvro~ CvzchL~~~cr +jso~oO~ov lcivqai~ Casrv.

...thus the fulfilment of what is potentially, as such, is motion.'*

Ross claims that this passage allows us to define change as the actualization of potential. He says that "alteration is the actualization of that which can be altered, growth and dimunition of that which can grow or diminish, generation and destruction of that which can be generated or destroyed, locomotion of that which can be moved in space."ll Since all of these involve motion (as AristotIe understands it) in some respect, it would seem that we require motion of some sort, though not local motion - i.e. change of place - in order for the process of actualization to work. If we say that geometrical objects can be generated, then according to this passage we can say that they are being actualized. This seems to indicate that we can take actuality and potentiality in stages (which agrees with the notion of potential 133 knowledge when asleep, when awake, and when used). Thus a line may exist only potentially, but be actualized by the movement of a point.I2

If we are actualizing geometrical objects, then there must be something in which they can exist potentially, as they clearly cannot exist potentially in a void. There are at least two possibilities. The first is that they exist only as potential parts of a continuous magnitude.

This couId be argued on the grounds that we arrive at the knowledge of geometrical objects by means of abstraction from regular objects. Thus, as we need the healthy in order to have health, we need these objects in order to have geometrical objects. This possibility seems needlessly restrictive. It seems to assume that we cannot consider the objects of geometry without some concrete object on which to base our . I believe that the more plausible position is to say that the objects of geometry exist in matter, but at a less concrete stage than the previous possibility. That is, what 1 am suggesting is that they exist, at least potentially, in intelligible matter. This would allow the geometer to consider the objects of geometry even in the absence of some solid from which to abstract. The mere presence of intelligible matter, which, if the earlier discussion of it makes sense, must exist in all places except those which are void, would then allow their contemplation.

Having made the assumption that points exist potentially in intelligible matter, we can now return to the idea that their actualization requires some sort of motion. The passage above relates the actualization of potentiality to motion. On a philological note, Ross tells us that

Bvzeki~~~amust here mean 'actualization', not 'actuality': it is the passage from potentiality to actuality that is ~ivqat~.'~ 134

Thus the actualization or fulfdment in question requires motion to be achieved, though it may not itself be a motion. In his commentary on the Metaphysica he differentiates between

BvrcAL~accand Bvdpy era. He says there that "Bvbpy ELCXmeans activity or actualization while kvtelbxem means the resulting actuality or perfe~tion."'~This fits with the etymology of the two words. ' Evdpys~cccomes from the combination of the prefix tv-, meaning "within," and tpyeiv, "arare active voiceof the common verb 6pydC~aBa~,which as a middle deponent has the active meaning of "to work," "to do, "to act," or "to be busy." l5

'EvteLE'~~ra,on the other hand, he says, comes from "bv TEAEL EXELV" meaning "to be complete." l6

Charlton claims that the 6dvap~<-Cvtde'~aadistinction is more accurately rendered as the distinction between possibility and fulfilment than between power and exercise." While part of Charlton's concern is with the connection between universal and particular, and this is what Witt's remarks on Charlton seems to take issue with'', there does not seem to be substantial disagreement that these terms can be used in the way Charlton proposes. Furthermore, both of these interpretations agree with Graham's assessment of

Kosman's view, namely that ~VTE~~XEI~signifies a product, not a proce~s.'~

Whatever it is, the process in question constitutes a motion which may be found by way of the connection made between geometry and potentiality as explained in the

Metap hys ica It is by actualization also that geometrical are discovered; for it is by dividing the given figures that people discover them. If they had been already divided, the relations would have been obvious; but as it is the divisions are present only potentially. Why are the angles of the triangle equal to two right angles? Because the angles about one point are equal to two right angles. If, then, the line parallel to the side had been already drawn, the theorem would have been evident to any one as soon as he saw the figure. Why is the angle in a semicircle in all cases a right angle? Because if three lines are equal - the two which form the base, and the perpendicular from the centre - the conclusion is evident at a glance to one who knows this premise.20

Of this passage Aquinas says

He [Aristotle] accordingly says, first that "geometricalconstructions," i.e., geometrical descriptions. "are discovered," i.e., made known by discovery in the actual drawing of the figures. For geometers discover the which they seek by dividing lines and surfaces. And division brings into actual existence the things which exist potentially; for the parts of a continuous whole are in the whole potentially before division takes place. However, if ail had been divided to the extent necessary for discovering the truth, the conclusions which are being sought would then be evident. But since divisions of this kind exist potentially in the fist drawing of geometrical figures, the ~thwhich is being sought does not therefore become evident immediately.21

This seems to imply that in order to discover anything about geometry, one must first have some concrete construction, perhaps an actual drawing if we are to take a strict reading of

Aquinas, and then use this to perform the relevant divisions to actualize the relevant potentials. Taken as a hard-line position, this seems to contradict the interpretation which

Ross gives. Ross tells us that this passage allows us to equate the thinking of the geometer to the actuality required to actualize the potential existence of the objects of geometry.22The geometer performs an action, i.e. thinking, which we might take to be a sort of motion. This constitutes the actuality required for our purposes. Thus, the potential existence of the geometrical object is actualized. The difficulty here lies in understanding what is meant by the tern "discovery." If we say that one can discover the of geometry by contemplation, then Ross's interpretation stands. It would imply that division in thought is adequate to actualize potentials to the degree necessary for understanding geometrical truths.

However, if the actualization can only be done by means of dividing a physical object or doing the equivalent on a diagram, then Aquinas's view must stand.

Ross gives us a reason to soften the hard-line position to one which is reconcilable with his position. He writes

What [Aristotle] says, then, is that 'geometrical constructions are discovered by an activity; for we find them by dividing'. The activity is later described as v6qo15, and this may not seem inconsistent with the description of it as a division. But it is not really so, for division here does not mean the drawing of lines with chalk or pen but the apprehension that the geometrical figures with which we are dealing are divisible in certain ways. The geometer is dealing with figures which are voqrd, and his essential activity is vdqol~, not the construction of anything aio0qr6v; the latter is merely an aid to the former.23

This tells us that we need not physically perform some division in order to gain knowledge of geometry. We need only apprehend the figures in question. Thus, thought is as valid a method for understanding geometry as sensory apprehension.

Further confirmation of the need for actualizing potentials can be found in Physica

@. Aristotle writes Perhaps there is nothing to prevent each of the two parts, or at any rate one of them, that which is moved, being potentially divided though actually undivided, so that if it is divided it will not continue in the possession of the same nature; and so there is nothing to prevent self-motion residing primarily in things that are potentially divisible.t4

Ross notes that Aristotle is discussing an drxopia about the divisibility of the moved and the mover. He asks

Is the answer that each of the two, or the moved part, may be potentially divisible, provided it is actually undivided, but that if it is divided, it no longer has the same nature; so that the power of causing motion or being moved may be in divisible things

Suppose that Ross is correct and that this is the answer. When we apply this to the problem of potency and geometry we find that it gives the answer which has already been put forward.

An object is divisible potentially at any location. Once a division has been made actually, the nature has changed. Thus the nature of the object has changed, and division is no longer possible in the same way as it was prior to division.

Leaving this matter for the moment, consider that once a division has been made in an object, there is a boundary between two parts. We can return to the question of what differentiates between a division in a continuous ubject and the division which differentiates between two contiguous objects. Clearly, if the two objects are flush, then it would seem initially that there is no difference between these two situations. This has formed part of the basis of the earlier discussion of boundaries. However, given the distinction between potency and actuality, we may have a resolution. It is hinted at in De Anima. L~ovtafiv ypccppat~mjv-d~oirepo~ 62 to6rov 06 tbv aGt6v tpdxov 6uvar6~kor~v, &M' 6 pdv cit~rl, ye'vo~ro~o8tov rai ti CAq, 6 6 ' dt~ pouAqBeis Guvat6~fkop~iv, Zv pj rL ~ob6ot~tQv Efoeev 6 6 * jGq 8eop&v, Bvrdq~i~Qv ud ~upiogtn~otdpevoc ~66~ rb A. drp46tepo~yBv ohoi irp3ror, ~arh~~VCY~IV t2motijpov~< C~VTES,

But now we must distinguish different senses in which things can be said to be potential or actual; at the moment we are speaking as if each of these phrases had only one sense. We can speak of something as a knower either when we say that man is a knower, meaning that man falls within the class of that know or have knowledge, or as when we are speaking of a man who possesses knowledge of grammar; each of these has potentiality, but not in the same way: the one because his kind or manner is such and such, the other because he can reflect when he wants to, if nothing external prevents him. And there is the man who is already reflecting - he is a knower in actuality and in the most proper sense is knowing, e.g. this A. Both the former are potential kfiowers, who realize their respective potentialities, the one by change of quality, i.e. repeated transitions from one state to its opposite under instruction, the other in another way by the transition from the inactive possession of sense or grammar to their active exercise?

What Aristotle is suggesting here is that actuality and potency are not merely opposite concepts. This is not to make the claim that there is a continuum of intermediate states between them, but rather that the relationship is more complex than a simple binary one.

Recall that there are two terms used to describe actuality - tvrebi~cuxand ivlpyam.

* Evteld~~~clis the resulting product. Evd py eta, however, is a process or activity. Thus, even what we claim is a result, and therefore an actuality, is still potential with respect to any further changes. Specifically, we have here the case of potential knowers. Thus, a person knows something potentially without having learned it simply by virtue of being a human with the capacity to learn. Once someone is taught some piece of knowledge, that person is 139 actually a knower and has the capacity to use the knowledge. However, if the knowledge is not being used, say, for example, that the person is asleep, that person can be thought of as a potential knower once again. This time, though. it is not in virtue of not possessing the knowledge. We have already made the claim that the knowledge is known, but only as capacity, not as expression. The potential is there because the knowledge is not being actively expressed. Thus, actualization reaches an extreme, we might say, when all potentials belonging to an object have been actualized.

Actualizin~Potentials

I have claimed that this notion of degrees of potentiality as it has been applied to knowledge can be applied to the continuous-contiguous problem. I feel that it can be done in the following way. Consider a continuous object. We know that it must be potentially divisible for Aristotle, as otherwise he would be forced to take an atomist position. Suppose that we mentally construct a division, but do not actually divide the object. Thus, we have a continuous object in which we have actualized a plane, albeit only mentally. The object has not been actually divided, but because we can envision such a division being made it is potentially divided. Suppose that we make our division, being careful not to lose any material and to keep the edges flush, leaving our chosen plane to act as the limit of both sides. Thus, we have the same situation, only the division has actually been made at the chosen plane. Unlike the earlier situation, we now have a state of affairs where we could potentially separate the two portions, changing the single boundary into two boundaries. We are not really dividing the boundary, as that would lead us to say that we can divide an object without width along its width, a clearly absurd concept. The situation is more akin to taking

two lines which have been in whole-with-whole contact along their lengths and separating

them. Essentially, what we are saying is that the difference between the two situations is that between two contiguous objects we have a boundary which may actually be one, but is potentially two. The situation is analogous to the potential knowers. The continuous body is like the potentid knower without the knowledge. The contiguous body is like the potential knower with knowledge. By performing successive actualizations we can eventually get from a single body to two distinct bodies. Similarly, we can perform the entire operation in reverse in order to get a single continuous body from two separate, but potentially continuous, bodies.

It is to be understood, despite the fact that there are degrees of actuality, that these degrees need not necessarily be actualized in a given object, though the actuality must exist in some way in some thing. We know the latter claim because of a later passage in De Anima where Aristotte tells us

Actual knowledge is identical with its object: potential knowledge in the individual is in time prior to actual knowledge but absolutely it has no priority even in time; for all things that come into being arise from what actually is.27

Thus, while the potential may be prior in time to the actual in a particular object, in the 'big picture' the actual always has priority. There are several reasons for the other claim, namely that the potential need not be actualized. Arisiotle writes

142

in an object. Since atomism implies just this sort of finite divisibility, it is easily seen how

this leads to doctrines such as atomism which Aristotle goes to such Lengths to refute.

There is yet another reason why objects cannot actually be made up of an infinite

number of objects. Consider that there are potentidly an infinite number of points, lines and

planes in any solid object. Aristotle tells us that if these had actual existence then division

would be impossible. In order to divide any solid, it would have to be cut at a plane, a plane

at a line, and a line at a point. But since points cannot be divided, neither can the more

complex entities? However, if these things exist only potentially, as already posited, then

they can exist in sensible objects.

Solving the Problems With Boundaries

With this understanding of how the actual and potential work, we can return to some

of the problems raised in the earlier chapters. In the discussion of boundaries a solution was

proposed to the problem of how it is that boundaries can be in contact. It was suggested that

we have touching when two types of matter come into contact but do not become like one another, and we have continuity when they do become like one another. Thus, when the

limits share enough properties, they allow us to say that the two touching objects are continuous. One of the examples given to illustrate this was the division of a geometrical

object. If we have an undivided square, we can say that the right half and the left half are

continuous. When we divide the square in two, but do not yet separate the halves, we have two shapes in contact at a boundary. When we separate the halves we have two objects in

succession. When this example was first mentioned, the term 'actualization' was thrown 143 around rather casually. It was said that by actualizing any division in a square we would have two distinct shapes in contact and not a single continuous shape. We can now understand more precisely what this means, so we can solve the problems with this solution which we were previously unable to handle.

When this soiution was proposed. I said that it was incomplete because it did not tell us how two objects could come to be in the same place at the same time without violating the rules of Aristotle's physics. Further, it only complicated one of the problems examined in a previous chapter, namely how points can be in contact. This is because this theory did not tell us how two points could be in whole-with-whole contact without mixing. The problem here is not so much that points are entities that should not be able to mix, but rather that if being in whole-with-whole contact involves mixing of some sort, then it seems we must say that when two objects touch at a point they are always continuous. This means that when we have two geometrical objects which are tangent, they must be continuous. Thus, it was suggested that what we need in order to solve this problem is a better understanding of mixing. Now, having examined mixing and, perhaps more importantly, actuality, potentiality, and intelligible matter, we can begin to fill in the gaps that worried us earlier.

Since it was suggested that the geometrical problem, and especially the problem of points in contact, was the major impediment to the problem, I will deal with it first.

Consider that an ideal point, that is to say a point as an object of geometry, is something that we have already said is a potential. It was claimed that points exist potentially in all places, but only exist actually once they have been selected. What needs to be solved here is the question of how it is that two objects can share a point, but still be contiguous, not continuous. One possible solution might come from Aristotle when he writes about time, the 'now' in particular.

Here, too, there is a correspondence with the point; for the point also both connects and terminates the length - it is the beginning of one and the end of another. But when you take it in this way, using the one point as two, a pause is necessary, if the same point is to be the beginning and the end?

This suggests that the solution is really just a matter of looking at things in the right way. If we pause when we consider a point then we can think of it as the end of one object. By continuing from this same point, we can then think of it as the start of a second object. Thus, we have contiguity. If we do not pause in this way, then we have continuity. As simplistic as this sounds, when we consider it in relation to the actual-potential distinction, it becomes significantly more complex.

It is interesting to note that the idea that one's analysis of a situation can change depending upon how one looks at it seems to have eluded Aristotle. This is most clearly evident in his famous remarks about the last moment of change.

But there are two ways of talking about that primarily in which something has changed. On the one hand it may mean the primary time at which the change is completed - the moment when it is correct to say 'it has changed'; on the other hand it may mean the primary time at which it began to change. Now the primary time that has reference to the end of the change is something really existent; for a change may be completed, and there is such thing as an end of change, which we have in fact shown to be indivisible because it is a limit. But that which has reference to the beginning is not existent at all; for there is no such thing as a beginning of change, nor any primary time at which it was changing.32

What we see here is Aristotle's assertion that there is no such thing as the first moment of change. The argument he presents rests upon the ciairn that if an object is changing, then it will be in motion. However, if there is a first moment of change, then we will have some time when the object is both at rest and in motion, both not changing and changing. Ross explains it this way:

No part of the process can be called absolutely fust, because the part may be divided again, thus revealing a prior first, and so ad infiniturn. Similarly of course no part of the process can strictly be called the end; but the limit (~8per5)exists not as a part of the process but as an indivisible limit to it."

What Aristotle seems to have missed is what Ross points out here. The analysis of change can be done from the other direction. By this I mean that we can say that if we look at the last moment of change, then we can say that there is no last moment of change in addition to saying that there is no first moment. I am unsure as to why Aristotle overlooked this possibility. I include it here, though, to show two things. First, it serves as another example of how our analysis of a situation depends upon how we look at the situation. In this case, if we hold either end of the process of change as an absolute then the other end becomes impossible to find. Second, it shows that, despite his best efforts, there are some problems that Aristotle had the tools to solve but which, for some reason, he did not tackle.

Returning to the proposed solution, when we examine a specific point in any given 146

object, we do not always look at this point in isolation. What I mean by this is that points

are usually identified in objects for the sake of some purpose. In the case of an object of

geometry, it is often to make some deduction about the properties of the object. Thus, we

identify a point in the centre of a circle in order to show that a point exists which is

equidistant from all points on the edge of the circle. When we treat physical objects as things

from which we might abstract. then the same sort of process applies. A ball, when thought

of as a sphere, might have points identified in order to illustrate some geometrical principle.

When we consider physical objects as things in themselves, and by this I mean that we do

not think of them as geometrical objects, we are not redly concerned with the points that

exist within them potentially. When we do consider the points which exist as part of the

objects, it is generally in some oblique manner. For example, suppose that a ball is on a

table. We have already seen that it is really only the geometer who will claim that the two

touch at a single point. When we analyze this situation in a geometrical sense, we say that the ball, at least an ideal version of the bail, touches the flat surface at a single point.

However, the physicist, who deals with the objects as sensible and not as idealized versions, will not make such a ciaim. This suggests that the physicist will say that there is a line or plane where the two objects are in contact. While points may potentialIy exist in lines or planes, the physicist, and by extension anyone who considers an object as sensible, need only consider the boundaries which can actually be sensed. Thus, while points may be at issue

when the boundaries themselves are examined, they are of secondary concern when sensible objects are examined.

The reason I make the distinction between the ways we deal with points is that these 147 distinctions affect how we actualize points. We have seen that Aristotle tells us that points can act both as connections and terminations. Thus, when I examine a geometrical object such as a Line, I might choose to examine a point on that line as a termination of one part and the start of another. On the other hand, I might choose to see it as the thing which connects one line to another. Think of the situation as follows. Suppose we have two people, my sister and myself, who are on a train going from Toronto to Windsor. The train stops in

London, my sister disembarks, and my brother boards. From the point of view of my sister, this is the end of a trip. For my brother, it is the start of a trip. From my vantage point, as one who has not left the train but has merely ceased moving for a brief time, this is merely a stop which connects the first part of the trip to the second part. Although there is only one stop, each rider can see it in a different way, and none is wrong. Likewise, I may actualize a point as the termination of one line and the start of another, or I might actualize it as part of the line. Neither view is wrong. It all depends upon the purpose behind the actualization.

Ross does not seem to agree with this position. In his discussion of Zeno's paradoxes, he finds that one of the solutions presented by Aristotle to solve the Dichotomy paradox is implausible. The solution presented is very much like the actuaVpotentia1 solution I have just given. When one moves along a line, the points on the line are actual only in certain cases.

For example, the points in the middle of a line are potential, whereas the points at the limits are actual." Further, if we think of a line as actually being composed of an infinite number points, then it is impossible to traverse any distance. However, if we maintain that the points of which a line is composed are only potential, then we can move along the line. Aristotle is maintaining that there are only an infinite number of points in a qualified sense, namely 148 that they are only potential. In any other sense there is not an infinite number of points.35

Ross's difficulty with this solution is as follows.

It surely cannot be maintained that a moving actualizes a point of space by coming to rest at it. It can come to rest only at a point that is there to be rested at. And when it does not rest but moves continuously, the pre- existence of the points on its course is equally pre-supposed by its passage through them.16

It seems that the problem Ross has with Aristotle's solution is based upon a reluctance to accept the idea that points can exist potentially and be actualized by certain actions - coming to rest at or counting a point, for example. However, beyond his assertion that this 'surely cannot be maintained, I can find no substantial objection on Ross' part to show why this is not possible. While it may be the case that this does not adequately solve the paradox - and this is certainly a topic for debate - it does not seem to me that Ross has any substantial reason for rejecting this form of solution other than the preferences he expresses for the solutions given by Cantor and ~usse11.~~

Returning to the earlier idea about the relation of continuity and contiguity to actuality and potentiaiity, we can now see how the connection can be made. When 1 actualize a point in a continuous object, say a line, 1 can consider it to either make the parts of the line continuous or contiguous. This is so because, as we have seen, the continuous is a subdivision of the contiguous. Thus, when I actualize the point it can be either as a connection between two separate parts, thereby acting as that which holds the two together, or it can be as the termination of one part and the start of another. However, when I have two objects which arecontiguous but not continuous, such as a ball on a table, the point can only be actualized as a termination and not as a connection. As mentioned earlier, it comes down to a matter of perspective. In some cases, we can see points as connections, in others perhaps only as terminations.

It is interesting to note that this solution might tell us how we are to understand a remarkable passage at the end of De Lineis Insecabilibus which introduces an altogether new geometrical entity, the joint.

Again, a point is not an indivisible joint. For a joint is always the limit of two things, but a point is the limit of one line. Moreover a point is a limit, but a joint is more of the nature of a division. Again, a line and a plane will be joints; for they are analogous to the point. Again, a joint is in a sense on account of movement (which explains the verse of Empedocles 'a joint binds two'); but a point is found also in immovable things. Again, nobody has an infinity of joints in his body or his hand, but he has an infinity of points. Moreover, there is no joint of a stone, nor has it any; but it has points.38

We are told here that points and joints are two different things for various reasons. Among these is the idea that a joint divides two things, whereas a point is the limit of one thing.

However, when a closer look is taken, it does not seem to be the case that what is being said is that points cannot be joints. Rather, the suggestion is that a point is not the same thing as a joint in the sense that sometimes points are not joints. This might allow us to say that a point can be a joint when it is taken as the connection between two lines, but that when 150 understood as a termination, a point is not a joint.

The distinction between points and joints in this passage is extremely interesting. So far as 1have been able to tell, this passage is the only one in all of Aristotle's works that deals with joints (6p8pcc) as geometrical objects and not as parts of a body. Generally, when joints are mentioned, they involve animals in some way. In fact, mentions of joints generally occur in the biological works, such as De Motu ~nimalium.~~where they are discussed as simply another part of the body. This passage gives us an indication that joints were also accorded geometrical status by some. While it is clear from the passage that Aristotle did not think of points in this way, it is clear that someone who was known to him and the members of his school gave such a description, though it is unknown who did so? Whoever it was, the definition must have been accepted enough to justify being attacked in this work. This gives us still more reason to think that a commonly accepted understanding of points was not yet available to Aristotle and that this was a topic of debate among scholars of his time.

Despite the uniqueness of this passage, since we know that Aristotle himself did not make the claim that points were indivisible joints, we can return to his view and the solutions we had found. It seems clear that the actuaVpotentia1 distinction solves the problem of points. However, it still needs to be clarified exactly how the equivdent higher-dimensional problem is solved. Specifically, we need to see how actualization helps to solve the problem of contact and mixing at boundaries.

Solving the Problems of Coincident Boundaries, Instantaneous Change, and Generation

The problems mentioned in the last chapter can now be dealt with. Consider first the 151

problem of coincident boundaries. It was suggested there that there is a problem when we

consider that when two boundaries touch, they mix. This was problematic since there would

be no way to differentiate between a single body and two different bodies. However, we now

have a way to do so. If we say that the boundaries have touched, then we must either have

continuity or contiguity. The way to determine which we have is to note the degree of

mixing. If the boundaries are mixed to the degree that they are actually one. just as the

boundary of a drop of water becomes one with the boundary of a greater pool of water, then

we have continuity. However, if the two boundaries mix only to the degree such that they

are only potentially one, then we have a situation where we have only contiguity.

This solution has the added advantage of allowing us to solve the problem of generation. Recall that this problem was that when two bodies touch, if they mix then we

may have a third type of matter between the two original types. If we say that contiguity allows for mixing such that the items are potentially one, but not actually so, then we might say that there is no longer a problem. A brick of tin and a brick of copper may be contiguous, but so long as they are only potentially mixed, they do not actually form bronze at the boundary. The boundary is both copper and tin, but requires something more before it becomes bronze. Perhaps an easier way to look at this is with a problem from an even earlier chapter.

The idea that boundaries behave in this way is not entirely unanticipated. In his discussion of change, we see that Aristotle uses the notion of intelligibility in order to explain how contraries can act upon one another. He claims that contraries do not affect one another per se, but rather they affect some substrat~m.~'Likewise, when we look at boundaries, we might say that we are not simply looking at two entirely different substances. Rather, we might say instead that we are looking at one type of matter - intelligible matter - acting as a substratum with various properties superimposed upon it. Ackrill notes that

[Aristotle] thought that one element could change into another, by a change in one of the characteristics; the hot-dry element, for example, would change into the cold-dry element if it lost heat and became cold. But now, if these changes are to be possible, each element must itself be a compound of 'prime matter', matter with no characteris tics, and two of the basic ~haracteristics.~~

This concept of prime matter is the same as the concept of intelligible matter that we have been using. Thus, we have a situation where we can have boundaries between any two types of matter explained by using the idea of the properties of the matter in question being held by a substratum of intelligible matter.

In the discussion of boundaries, the case of a drop of wine being added to a body of water was examined. One of the difficuities mentioned there was that Aristotle tells us that the wine will become part of the water, but does not go further. We were left to wonder when the boundary between wine and water stops existing, as well as wondering how any such change could happen instantaneously. The answer, if this theory works, would run as follows. When the wine first touches the water, the boundaries are potentially one, but actually two. After this stage, it is possible for the potential to be actualized such that the two liquids become one. This allows for the transition from wine to water to take place over time, and explains away the idea that the change happens instantaneously. This still requires some fine tuning. Ross describes the process of change in the following way.

m, which is actually x and potentially y, imparts x-ness to n, which is actually y and potentially x, while n simultaneously imparts y-ness to m? 153

This suggests that while the wine is turning into water, the water is turning into wine. We might solve this by turning again to intelligible matter and saying that we have certain properties retained by the matter while others are negated or diminished. In fact, by saying that the properties of the wine as they exist as part of intelligible matter are diminished we avoid another problem of mixing. Thus, for example, if we say that the drop of wine becomes water in an unqualified sense, then we might fall into the trap of having to say that one can add as much wine to water as one wants and still have water provided the wine is always added one drop at a time. By using the idea of properties in intelligible matter, we can say that with every drop added the properties of the water change ever so slightly towards being like the properties of the added wine. Thus, we will eventually have watery wine rather than mere water.

Let us review what has been accomplished so far in this chapter. It began by explaining the concepts of actuality and potentiality so that they could be used to help solve earlier problems. These concepts were then related to intelligible matter and mixing.

Forging this relation allowed us to address the problems we had with points and boundaries and to reevaluate the solutions presented in the earlier chapters. Having done this, we are now in a position to solve the problems which opened this work. This is what will be undertaken in the last chapter.

1. Metaphysica 1041a32-b9.

2. Jonathan Barnes, "Metaphysics " , The Cambridge Companion to Aristotle, ed. Jonathan Barnes (Cambridge University Press, 1996) 97. 3. Melbourne G. Evans, The Physical Philosophy ofAristotle (Albuquerque, The University of New Mexico Press, 1964) 26.

4. Metaphysica 1048a30-35.

5. De Generatione Animalium 734b19-22.

6. Metaphysica 1O49aS- 12.

7. W. D. Ross, Aristotle's Metaphysics (volume 2) (Oxford, Clarendon Press, 1953) 256.

8. Metaphysica 1002a20-28.

9. St. Thomas Aquinas, Commentary of the Metaphysics of Aristotle, volume 2, trans. John P. Rowan (Chicago, Henry Regency Company, 1961) 680.

11. W. D. Ross, Aristotle's Physics (Oxford, Clarendon Press, 1955) 359.

12. ibid. 536.

13. ibid. 537.

14. Ross, Aristotle's Metaphysics, vol. 2,245.

15. George A. Blair, Energia and Entelecheia: "Act" in Aristotle (University of Ottawa Press, 1992) 17.

16. ibid. 79.

17. William Charlton, "Aristotle and the Uses of Actuality," Proceedings of the Boston Area Colloquium in Ancient Philosophy 5 (1989):18.

18. Charlotte Witt, "Commentaryon Charlton," Proceedings of the Boston Area Colloquium in Ancient Philosophy 5 (1989): 23-26.

19. Daniel W. Graham, "Aristotle's Definition of Motion", Ancient Philosophy 8 (198 8): 209.

20. Metaphysica 1051 a2 1-29.

21. Aquinas, vol. 2, 696.

22. Ross, Aristotle's Metaphysics, vol. 2, 267. 23. ibid. 268-269.

24. Physica 258a32-b4.

25. Ross, Aristotle 's Physics 440.

28. Physica 206a14-18.

29. Ross, Aristotle's Physics 555.

3 1. Physica 220a9- 13.

32. Physica 236a7-15.

33. Ross, Aristotle's Physics 649-650.

34. Physica 262b3 1.

35. Physica 263b2-8.

36. Ross, Aristotle S Physics 74-75.

37. Ross, Aristotle S Physics 75.

38. De Lineis Insecabilibus 972b25-34.

39. De Motu Animalium 698a16ff.

40. H. H. Joachim, De Lineis Insecabilibus, in The Works of Aristotle Translated Into English: Vol. VI. Opuscula, ed. W. D. Ross (Oxford, Clarendon Press, 1913), note on 972b25.

41. Physica 189b6.

42. J. L. Ackrill, Aristotle the Philosopher (Oxford,Clarendon Press, 1994) 32-33.

43. Ross, Aristotle's Physics 45. Cha~ter6 - Conclusion

Review - What Has Been Uncovered Thus Far This section sums up what has been accomplished to this point. The various explorations made in each chapter are put into the larger context of the problem of touching in order to make clearer how the entire project fits together.

Solving-the Original Problems We return to the three basic problems mentioned in the introduction - geometrical tangency, touching, and physical tangency. Using the concepts examined in the intervening chapters, solutions are offered for ail of these problems.

The Significance of the Findings This section briefly discusses the importance of what has been accomplished. Also, it makes clear that these are not necessarily the solutions Aristode offered, but rather that they are solutions he could have offered while staying true to his general outlook on both geometrical and physical matters. 157

Review - What Has Been Uncovered Thus Far

A review of the various topics discussed up until now might be helpful. This is especially so because despite the attempt to impose order on the problems examined, they are still somewhat tangled. Therefore, I will attempt to untangle the threads, or at least loosen the knots in them, before returning to the problems which prompted the inquiry initially.

The first topic discussed was that of points. The reason given for starting there was that it was assumed that since the problems in the introduction dealt primarily with limits, an examination of the simplest type of limit would allow us to proceed. This, it turns out, was not entirely the case. I was able to proceed, but only after discovering that the concept of a point in the Corpus is not as simple as is generally thought. Perhaps the most important discovery in that chapter was that AristotIe did not see points merely as geometrical ideals, but rather both as geometrical entities and as sized particles akin to atoms. When thought of as purely geometrical, however, an advance was made. Namely, we saw that tangency is possible when points are in whole-with-whole contact, though this is not unproblematic.

Questions such as how to differentiate between two tangent objects were still very much unsolved. This was especially so with ideal points, given that they have no parts by which they cm be differentiated. Thus, when in contact it would appear that we have only one point. The result of the analysis was that points could not give us the required solutions.

The next step was to apply the results obtained from the examination of points to more complex geometrical entities such as lines, planes, and physical boundaries. Such an analysis, if successful, would have allowed us to immediately solve the problems presented 158 in the introduction. This is because those problems are all essentially problems dealing with boundaries in contact. However, as with the chapter on points, we ran into difficulties centred on the problem of two entities occupying the same place. There were two problems which had to be dealt with in order to solve these difficulties. One ofthese was the problem of mixing. Since we were concerned with what would happen when two objects occupied the same place, we needed a coherent theory of mixing, which had not been presented. The other problem was that we needed to establish a stronger link between geometry and physics.

This would then allow us to have a single theory of touching which could be applied universally, rather than having one theory which would apply to physical objects and another which would apply to mathematical objects.

The next two chapters provided the necessary answers. The introduction of three important related concepts - mixing, intelligible matter, and potency - gave the required answers needed to solve the puzzles we were left with in the first two chapters. More accurately, the problems associated with mixing were solved by the concepts of intelligible matter and actuality. With these ideas firmly in place, the difficulties encountered in the earlier chapters could be returned to. The problems associated with both boundaries and points were solved by using these notions, thus paving the way for us to be able to solve the initial problems.

Solving- the Original Problems

There were four main problems set out in the introduction. In the order they were presented they were 1) the problem of touching when applied to geometrical objects tangent to one another 2) the problem of touching when applied to physical objects tangent to one another 3) the problem of touching at boundaries greater than a single point 4) the problem of touching when applied to geometrical situations abstracted from the equivalent physical situation

With the results obtained in the previous chapters, we can now return to each of these problems, though not in the order in which they were initially presented, and offer solutions that should alleviate them.

1. Basic Tangency

When first presented, tangency was dismissed fairly quickly as not being troublesome. It was suggested that when we say two geometrical objects are tangent to one another, they are standardly thought of as sharing a single point. While this may be so, now that an in-depth analysis of points has been done we should really return to this problem and see more precisely why this is so.

One way to think of the situation, the way it was presented in the introduction, is to think of two circles touching at a single point. Of course, we want to say that these two distinct circles are touching, as opposed to saying that there is a single continuous figure. In order to do this we must understand what happens at the point of tangency. Given the findings in the chapter on points, we might be inclined to say that although the situation seems to be one in which we have two objects in contact at a single point, the is rather different. In fact, what we have is two points on two distinct circles, each of which is in whole-with-whole contact with the other. However, such an analysis does not tell the whole 160 story. Recall that in the chapter on points, we found that this solution taken by itself is still problematic. We need to understand why we are able to differentiate between the circles and not simply say that they compose a single object.

The solution to this was found when we examined the concepts of actuality and potentiality. There we saw that when we consider a continuous geometrical object, we are able to understand any particular point as being both a connection between parts and a terminus. Thus, when we examine the geometrical situation involving tangent circles, we have a situation that is slightly more complex than we initially thought, but one which gives us the desired solution. The circles can be thought of as either a single figure or as two touching figures. Our understanding of which we have is determined by how we look at the point of tangency. If we say that it connects the two circles, we have the former situation.

If, rather, we understand that the point serves as the end of the radius of one and the beginning of the other, than we have the latter situation. Thus, we have a solution which will allow us to understand geometrical tangency without difficulty.

2. Touching at Boundaries

The third of the four problems dealt with the problems encountered when we have contact at a boundary that was not necessarily limited to a single point. When dealt with as a purely geometrical problem, this can be solved in the same manner as the earlier problem of tangency. That is, we can understand the shared limit to be a single boundary between two distinct parts of an object, or we can think of it as the end of one part and the beginning of another. The solution is essentially comes down to how we apprehend the boundary. 161

The more complex difficulty deals with the physical version of the problem. While we might be comfortable claiming that boundaries act in different ways in geometrical cases depending upon how we think of them, we are not as comfortable saying this in the physical case. The reason for this is that we would quickly find ourselves in a paradoxical situation were we to do so. Recall that an integral part of the physical problem deals with mixing at boundaries. We must concern ourselves with understanding how two objects can share a limit when they touch and yet not become a single object. This is an added complexity to the geometrical problem where the objects could be thought of as one object or two depending on our needs.

The solution to this problem is to be found by means of the concept of mixing and, as with the problems of basic tangency, the concepts of actuality and potentiality. When two objects are put together in such a way as to become continuous, they mix and basically lose any boundaries that were between them. However, when they are merely in contact the boundary remains. This is accomplished, as was suggested in the previous chapter, by the two objects in contact being mixed only in potentia. Thus, there remains a boundary between the objects, though it acts as a single object and not as two separate limits. Also, since the mixing has only happened potentially, we can say that there has not been any new material generated at the limit. For such an event to occur, mixing would actually have to take place. For the same reason - and by this I mean to suggest that it is because of the mixing taking place only potentidly - there is no need to worry about an object in contact being adulterated by the material with which it is in contact. Since no mixing has actually taken place the objects can remain pure. 162

3. Phvsical Tangency and Abstraction

The solution to the problem of how sensible objects come into contact also helps to give us the solution to the second of the four difficulties. Using this solution we are able to understand how two physical objects may come into contact with one another without each compting the other. In order to solve the problem of physical tangency there remains only to take the limiting case of a physical boundary. Theoretically, since we are working within the confines of a continuum theory, we could have objects which have perfect geometrical shapes. This is unlike an atomist theory where we are limited to approximations based upon the shapes and sizes of the atoms. Thus, if we have perfectly round spheres, we can simply apply the solution we found for boundaries to the points at which the spheres touch.

However, as we have established in the chapter on geometrical objects, it is not necessary for perfect geometrical objects to exist also as sensible ones. Geometrical objects are often simply tools for geometers. Thus, even when it appears to the ~iakedeye that we have contact between objects at a single point, we may actually have contact over a very tiny - but still finitely large - area. In this case, the solution we have already seen for contact between physical objects car, be applied.

Understanding this also allows us to understand how it is that we can abstract to the geometrical from the physical. First we must realize that the objects we examine when we do geometry are found by examining physical objects. However, the geometer is able to ignore the imperfections that may be present in the sensible object in order better to analyze the ideal geometrical equivalent. Thus, when have a situation where we have two sensible spheres which are in contact, the geometer can abstract away from all the sensible qualities 163 and correct for any imperfections. If the spheres are not perfectly shaped and are in contact over some small area, the geometer can smooth away the imperfections in the model and consider the situation as it would be if there were in fact perfectly shapzd spheres in contact.

The Significance of the Findinns

Having solved the puzzles presented at the sta~?of this work, I think it is important to make a brief statement about the importance of what has and has not been accomplished.

One thing that must be made clear is that I have no illusions that AristotIe actually made such explicit claims about objects in contact. Nor do I claim that what I have presented comprises the solutions which Aristotle himself gave - or even necessarily would have given - to these problems. It is clear that, though he did address related problems, for the most part he did not address these difficulties. What I have presented, then, is a way in which Aristotle could have used his theories of geometry and the continuum in order to solve problems of this nature.

This brings us back to Sharvy's remark, mentioned in the introduction. In his view, the problems associated with boundaries are easily solved by simply thinking of points as limits. Anytlung more complicated is not woah caring about. However, it is clear that with a little thought a much more satisfying result than Sharvy's solution to the problem can be found. While it is true that Shany was on the right track in thinking of points as limits, there is, as I have shown, certainly more to a complete solution. Furthermore, this solution does not necessitate that we interpret Aristotle as thinking about points as Descartes did. Rather, it allows us to see Aristotle as being unsettled about the solutions he presented. This, in turn, 164 allows us to say that he did not think of such things as points and boundaries with the clarity of those who came after him, but rather that he struggled with these concepts and often had more than one possible way of thinking about them. Thus, one virtue of this solution is that it allows us to view Aristotie's thought with a greater sense of completeness than we had before.

Something else of significance is Iess evident from the solutions I have presented, but is perhaps even more important in some ways. We know that Aristotle held his theory of the continuum in high regard. However, when we look at the application of this theory throughout the Corpus we find that there are discrepancies. This can be attributed in part to the fact that Aristotle wrote his works over a period of many years and it is natural to assume that he changed his mind or found new answers over those years. Add to this the pseudonymous nature of certain works, such as De Lineis Insecabifibus, and it is easy to see why it is often so difficult to pin Aristotle or the Aristotlelian school down on certain subjects. Nonetheless, what I have shown in this work is that even when we take these discrepancies into account we are able to form a more or Iess coherent picture of AristotIe's thought about a certain aspect of the continuum. This does not mean that the picture is entirely clear. The various topics examined have shown that Aristotle himself was often unclear as to the nature of those things he was examining.

The notion of having a clear idea of what was in Aristotle's mind is probably the most important part of this work. I am not saying here that what I have presented constitutes exactly what was in his mind. However, as I have often mentioned, I have attempted to go beyond those writers who make the claim that Aristotle's ideas of continuity, geometry, etc., 165 were essentially those that we hold. I have tried to put Aristotle's writings into his context.

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